Integration is one of the most important topics in calculus, but some integrals cannot be solved easily with basic integration rules alone. When an integral contains a product of two functions, integration by parts, also called partial integration, is often an effective method. The Partial Integration Calculator helps you work through this process by identifying the main components of the integration-by-parts formula and presenting the resulting expression in a clear format.

The calculator is designed for problems involving a product such as a polynomial multiplied by a trigonometric, exponential, or other supported function. Instead of manually remembering every step, you can enter the first function, the derivative portion of the second function, and the variable. The calculator then identifies u, dv, du, and v, and displays the integration-by-parts setup.

The standard formula is:$$\int u\,dv = uv-\int v\,du$$

This formula changes a difficult integral into another integral that may be easier to solve. Choosing the right function as u is an important part of the process.

Whether you are a student learning calculus, preparing for an exam, checking homework, or reviewing integration techniques, this Partial Integration Calculator can help you understand how the method works.

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What Is Partial Integration?

Partial integration is another name commonly used for integration by parts. It is a calculus technique used to integrate the product of two functions.

Suppose an integral has the general form:$$\int f(x)g'(x)\,dx$$

You can identify:$$u=f(x)$$

and$$dv=g'(x)\,dx$$

Then differentiate $u$:$$du=f'(x)\,dx$$

and integrate $dv$:$$v=g(x)$$

The original integral can then be rewritten using:$$\int u\,dv=uv-\int v\,du$$

The objective is usually to choose $u$ and $dv$ so that the remaining integral is simpler than the original one.

For example:$$\int x\cos(x)\,dx$$

can be treated as:$$u=x$$

and$$dv=\cos(x)\,dx$$

Therefore:$$du=dx$$

and:$$v=\sin(x)$$

Using integration by parts:$$\int x\cos(x)\,dx=x\sin(x)-\int\sin(x)\,dx$$

which becomes:$$x\sin(x)+\cos(x)+C$$

The calculator helps organize these intermediate steps.

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What Does the Partial Integration Calculator Do?

The Partial Integration Calculator accepts three primary pieces of information:

1. Function f(x) — the function you want to select as $u$.
2. Function g'(x) — the function multiplied by $dx$, which becomes $dv$.
3. Variable — the variable with respect to which the integration is performed.

After you click Calculate, the tool displays:

• Selected $u$
• $dv$
• $du$
• $v$
• The integration-by-parts formula
• The resulting integration-by-parts expression

This makes the calculator particularly useful for understanding the structure of integration by parts rather than simply receiving a final answer.

The tool supports several common functions, including powers of the variable, constants, sine, cosine, exponential functions, and natural logarithms.

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How to Use the Partial Integration Calculator

Using the calculator is straightforward. Follow these steps.

Step 1: Enter Function f(x)

In the Function f(x) field, enter the function you want to use as $u$.

Examples include:

• x
• x^2
• x^3
• sin(x)
• cos(x)
• e^x
• ln(x)
• 5

For example, if you are solving:$$\int x\cos(x)\,dx$$

enter:

x

as the first function.

Step 2: Enter Function g'(x)

Enter the second part of the integral in the Function g'(x) field.

For the example:$$\int x\cos(x)\,dx$$

enter:

cos(x)

The calculator treats this as the differential component:$$dv=\cos(x)\,dx$$

Step 3: Enter the Variable

The default variable is x.

For most standard calculus exercises, you can leave this field as x.

The calculator expects a single letter as the variable, such as:

• x
• y
• t
• z

Step 4: Click Calculate

Click the Calculate button.

The calculator processes the selected functions and shows the integration-by-parts components.

Step 5: Review the Results

The results include:

Selected u: the first function.

dv: the second function multiplied by the differential.

du: the derivative of $u$.

v: the antiderivative of $dv$.

The calculator also displays the formula:$$\int u\,dv=uv-\int v\,du$$

Finally, it provides the integration setup so you can continue solving the remaining integral.

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Partial Integration Formula Explained

The central formula used by the calculator is:$$\boxed{\int u\,dv=uv-\int v\,du}$$

Each part has a specific purpose.

u

The term $u$ is the function selected from the original product.

For example:$$u=x$$

dv

The term $dv$ is the remaining part of the integral together with $dx$.

If the integral is:$$\int x\cos(x)\,dx$$

you could select:$$dv=\cos(x)\,dx$$

du

The term $du$ is found by differentiating $u$.

If:$$u=x$$

then:$$du=dx$$

because:$$\frac{d}{dx}(x)=1$$

v

The term $v$ is found by integrating $dv$.

If:$$dv=\cos(x)\,dx$$

then:$$v=\sin(x)$$

because:$$\int\cos(x)\,dx=\sin(x)$$

Once these values are known, substitute them into:$$\int u\,dv=uv-\int v\,du$$

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The General Process of Integration by Parts

A useful way to remember the method is to follow four steps:

1. Choose u

Select one part of the integrand as $u$.

2. Find du

Differentiate $u$ to obtain $du$.

3. Choose dv

The remaining part of the integrand becomes $dv$.

4. Find v

Integrate $dv$ to obtain $v$.

Then substitute everything into:$$\int u\,dv=uv-\int v\,du$$

The ideal selection makes the remaining integral easier.

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Practical Example: ∫ x cos(x) dx

Consider:$$\int x\cos(x)\,dx$$

This is a classic integration-by-parts problem because it contains the product of a polynomial and a trigonometric function.

Choose:$$u=x$$

and:$$dv=\cos(x)\,dx$$

Differentiate $u$:$$du=dx$$

Integrate $dv$:$$v=\sin(x)$$

Now use the formula:$$\int u\,dv=uv-\int v\,du$$

Substitution gives:$$\int x\cos(x)\,dx = x\sin(x)-\int\sin(x)\,dx$$

Since:$$\int\sin(x)\,dx=-\cos(x)$$

we get:$$x\sin(x)+\cos(x)+C$$

Therefore:$$\boxed{\int x\cos(x)\,dx=x\sin(x)+\cos(x)+C}$$

When using the calculator, enter:

• Function f(x): x
• Function g'(x): cos(x)
• Variable: x

The tool identifies $u=x$, $du=1$, and $v=\sin(x)$, then presents the integration-by-parts expression.

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Example 2: ∫ x e^x dx

Consider another common problem:$$\int xe^x\,dx$$

Choose:$$u=x$$

and:$$dv=e^x\,dx$$

Differentiate:$$du=dx$$

Integrate:$$v=e^x$$

Apply the formula:$$\int xe^x\,dx=xe^x-\int e^x\,dx$$

Therefore:$$\int xe^x\,dx=xe^x-e^x+C$$

This can also be written as:$$\boxed{e^x(x-1)+C}$$

The calculator can help establish the integration-by-parts setup for this type of problem.

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Example 3: ∫ x² sin(x) dx

Consider:$$\int x^2\sin(x)\,dx$$

Select:$$u=x^2$$

and:$$dv=\sin(x)\,dx$$

Then:$$du=2x\,dx$$

and:$$v=-\cos(x)$$

Therefore:$$\int x^2\sin(x)\,dx = -x^2\cos(x)+\int2x\cos(x)\,dx$$

The remaining integral still contains a product, so integration by parts may need to be applied again.

This demonstrates an important point: integration by parts does not always solve the entire problem in one step. Sometimes the resulting integral requires another application of the same technique.

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Choosing the Right Function for u

One of the most important skills in integration by parts is selecting $u$.

A commonly taught guideline is LIATE, which helps prioritize the type of function that should usually be selected as $u$.

LIATE stands for:

• L — Logarithmic
• I — Inverse trigonometric
• A — Algebraic
• T — Trigonometric
• E — Exponential

For example:$$\int x\cos(x)\,dx$$

contains an algebraic function $x$ and a trigonometric function $\cos(x)$.

The algebraic function generally makes a good choice for $u$:$$u=x$$

For:$$\int x e^x\,dx$$

the algebraic function $x$ is typically selected as $u$, while $e^x$ becomes $dv$.

LIATE is a useful guideline rather than an absolute law. Some integrals require experimentation or a different strategy.

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Functions Supported by the Calculator

The calculator is designed around several common elementary functions.

Variable

The basic variable, such as:$$x$$

Powers

Examples include:$$x^2,\quad x^3,\quad x^4$$

The power rule for differentiation is:$$\frac{d}{dx}x^n=nx^{n-1}$$

Sine

$$\frac{d}{dx}\sin(x)=\cos(x)$$

and:$$\int\sin(x)\,dx=-\cos(x)$$

Cosine

$$\frac{d}{dx}\cos(x)=-\sin(x)$$

and:$$\int\cos(x)\,dx=\sin(x)$$

Exponential Function

For:$$e^x$$

both differentiation and integration produce:$$e^x$$

Natural Logarithm

For:$$\ln(x)$$

the derivative is:$$\frac{1}{x}$$

and:$$\int\frac{1}{x}\,dx=\ln(x)+C$$

Constants

A constant such as 5 has derivative zero:$$\frac{d}{dx}(5)=0$$

and:$$\int5\,dx=5x+C$$

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Why Use a Partial Integration Calculator?

There are several reasons a calculator like this can be useful.

1. Saves Time

Manually identifying $u$, $dv$, $du$, and $v$ for multiple practice problems can take time. A calculator provides the setup quickly.

2. Helps With Learning

The tool does not simply focus on a final expression. It displays the intermediate components, making it useful for students learning integration by parts.

3. Reduces Setup Errors

A common mistake is forgetting to differentiate $u$, integrate $dv$, or use the correct negative sign. Seeing each component separately makes the structure easier to check.

4. Useful for Homework Checking

Students can solve a problem independently and then use the calculator to compare their choice of $u$, $du$, and $v$.

5. Supports Different Common Functions

The calculator works with several frequently encountered functions, including powers, trigonometric functions, exponential functions, logarithms, and constants.

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When Should You Use Integration by Parts?

Integration by parts is especially useful when the integrand is a product of two functions.

Common examples include:$$x\sin(x)$$$$x\cos(x)$$$$xe^x$$$$x^2e^x$$$$x\ln(x)$$

It can also be useful for certain logarithmic and inverse-trigonometric integrals.

However, not every integral involving two expressions should automatically be solved with integration by parts. Substitution, algebraic simplification, trigonometric identities, or other integration techniques may be more appropriate.

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Partial Integration vs. Substitution

Integration by parts and substitution are different techniques.

Substitution is often useful when an integral contains a function and its derivative.

For example:$$\int2x(x^2+1)^5\,dx$$

is naturally suited to substitution.

Integration by parts, on the other hand, is particularly useful for products such as:$$\int xe^x\,dx$$

Knowing which technique to use is an important calculus skill.

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Common Mistakes in Integration by Parts

Forgetting the Minus Sign

The formula is:$$\int u\,dv=uv-\int v\,du$$

The subtraction sign is essential.

Choosing an Unhelpful u

A poor choice of $u$ can make the remaining integral more difficult.

Forgetting dx

Remember that:$$dv=g'(x)\,dx$$

The differential is part of the integration expression.

Incorrectly Integrating dv

If:$$dv=\cos(x)\,dx$$

then:$$v=\sin(x)$$

not $\cos(x)$.

Forgetting the Constant of Integration

For indefinite integrals, the final answer should generally include:$$+C$$

The calculator’s displayed setup includes the integration constant in the resulting expression.

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Important Tips for Better Results

For the best experience with the calculator, keep your expressions in a simple standard form.

Examples of useful entries include:

• x
• x^2
• x^3
• sin(x)
• cos(x)
• e^x
• ln(x)
• 1
• 5
• 1/x

Make sure both function fields contain valid expressions before calculating.

Also remember that the calculator is primarily a partial integration setup tool. The remaining integral shown after applying the formula may sometimes require additional mathematical work.

For more complicated expressions outside the supported forms, a manual solution or a more advanced symbolic integration method may be necessary.

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How to Check Your Answer

After applying integration by parts, you can check the result by differentiating your final answer.

For example, suppose you obtain:$$x\sin(x)+\cos(x)+C$$

Differentiate:$$\frac{d}{dx}[x\sin(x)+\cos(x)]$$

Using the product rule:$$\sin(x)+x\cos(x)-\sin(x)$$

The sine terms cancel:$$x\cos(x)$$

which matches the original integrand.

This is an excellent way to verify an indefinite integral.

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Partial Integration Calculator for Students

For students, the biggest benefit of this tool is understanding the process instead of memorizing answers.

When practicing, try solving each problem manually first. Identify $u$, $dv$, $du$, and $v$, then compare your work with the calculator.

For example, for:$$\int x^2e^x\,dx$$

ask yourself:

1. Which function should be $u$?
2. What is $du$?
3. What should be $dv$?
4. What is $v$?
5. Does the remaining integral become easier?
6. Is another application of integration by parts necessary?

This approach develops stronger calculus skills than simply copying the displayed result.

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Partial Integration in Calculus and Real-World Mathematics

Integration by parts is not just an academic formula. Integration techniques are fundamental to many areas of mathematics, science, engineering, economics, and physics.

Integrals are used to calculate quantities such as:

• Areas under curves
• Accumulated quantities
• Work
• Physical displacement
• Probability distributions
• Continuous growth models
• Mathematical expectations
• Engineering quantities
• Differential-equation solutions

Integration by parts becomes especially valuable when mathematical models contain products of different functions.

For this reason, learning how to recognize when integration by parts is appropriate can be more valuable than simply memorizing the formula.

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Frequently Asked Questions

1. What is a Partial Integration Calculator?

A Partial Integration Calculator is a tool that helps organize integration-by-parts problems by identifying $u$, $dv$, $du$, and $v$, then displaying the resulting formula.

2. What is the formula for partial integration?

The standard formula is:$$\int u\,dv=uv-\int v\,du$$

It converts an integral into another expression that may be easier to evaluate.

3. Is partial integration the same as integration by parts?

Yes. “Partial integration” and “integration by parts” refer to the same fundamental calculus technique.

4. What should I enter as f(x)?

Enter the function you want to designate as $u$. For example, for $\int x\cos(x)\,dx$, you can enter x.

5. What should I enter as g'(x)?

Enter the function that will become $dv$. For $\int x\cos(x)\,dx$, enter cos(x).

6. What does du mean?

$du$ represents the differential obtained by differentiating $u$. If $u=x^2$, then:$$du=2x\,dx$$

7. What does v represent?

$v$ is the antiderivative of $dv$. If:$$dv=\cos(x)\,dx$$

then:$$v=\sin(x)$$

8. How do I choose u?

A common guideline is LIATE: logarithmic, inverse trigonometric, algebraic, trigonometric, and exponential functions. The function appearing earlier in this guideline often makes a useful choice for $u$.

9. Can the calculator solve every integration problem?

No. The calculator focuses on common functions supported by its calculation rules. More complicated expressions may require manual methods or other integration techniques.

10. Can I use a variable other than x?

Yes. The calculator allows a single-letter variable. You can use a variable such as $t$, $y$, or $z$, provided the expressions use the same variable.

11. Why does integration by parts have a minus sign?

The minus sign comes from rearranging the product rule for differentiation. Starting with:$$d(uv)=u\,dv+v\,du$$

and rearranging gives:$$u\,dv=d(uv)-v\,du$$

which leads to:$$\int u\,dv=uv-\int v\,du$$

12. Does integration by parts always finish the problem in one step?

No. Some problems require integration by parts more than once. For example, polynomial-exponential products can require repeated applications.

13. What is the purpose of the constant C?

For an indefinite integral, $C$ represents an arbitrary constant because differentiating any constant produces zero.

14. How can I check an integration answer?

Differentiate the final answer. If its derivative matches the original integrand, the result is consistent with the original integral.

15. Is the Partial Integration Calculator useful for learning?

Yes. It can be particularly helpful for learning because it separates the process into $u$, $dv$, $du$, and $v$, allowing students to see how the formula is constructed.

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Final Thoughts

The Partial Integration Calculator provides a convenient way to work with one of the most important techniques in integral calculus. By entering a first function, a second derivative function, and the variable, you can quickly identify the essential components needed for integration by parts.

The key formula to remember is:$$\boxed{\int u\,dv=uv-\int v\,du}$$

The real skill, however, is learning how to choose $u$ and (dv` so that the resulting integral becomes easier. Guidelines such as LIATE can help, but practice is ultimately the best way to develop good judgment.

Use the calculator to check your setup, study the relationship between differentiation and integration, and practice solving the remaining integral yourself. With regular practice, expressions involving polynomials, trigonometric functions, exponential functions, and logarithms become much easier to recognize and solve.

Whether you are preparing for a calculus exam, completing assignments, revising integration techniques, or simply exploring calculus concepts, this Partial Integration Calculator can serve as a useful learning companion.