The Formula for Integration by Parts

Oct 13, 2025 by windailys.com 134 views

Introduction

Integration by parts is a powerful technique used to evaluate definite integrals of the form $\int f(x)g'(x) \, dx$ , where $f(x)$ and $g(x)$ are functions of $x$ . This method is particularly useful when the integral cannot be evaluated directly using basic integration rules. In this article, we will demonstrate how to use integration by parts to evaluate the definite integral $\int_1^2 x e^x \, dx$ .

The Formula for Integration by Parts

The formula for integration by parts is given by:

\int f(x)g'(x) \, dx = f(x)g(x) - \int f'(x)g(x) \, dx

where $f(x)$ and $g(x)$ are functions of $x$ . This formula allows us to integrate the product of two functions by differentiating one function and integrating the other.

Applying Integration by Parts to the Given Integral

To evaluate the definite integral $\int_1^2 x e^x \, dx$ , we can use integration by parts with $f(x) = x$ and $g(x) = e^x$ . We have:

\int_1^2 x e^x \, dx = \int_1^2 x e^x \, dx = \left[ x e^x \right]_1^2 - \int_1^2 e^x \, dx

Evaluating the First Term

The first term in the above equation is $x e^x$ . We can evaluate this term by substituting the limits of integration:

\left[ x e^x \right]_1^2 = 2 e^2 - 1 e^1

Evaluating the Second Term

The second term in the above equation is $\int_1^2 e^x \, dx$ . We can evaluate this term by using the formula for the integral of $e^x$ :

\int e^x \, dx = e^x + C

Therefore, we have:

\int_1^2 e^x \, dx = \left[ e^x \right]_1^2 = e^2 - e^1

Combining the Results

Now that we have evaluated both terms, we can combine the results to obtain the final answer:

\int_1^2 x e^x \, dx = 2 e^2 - 1 e^1 - (e^2 - e^1)

Simplifying the above equation, we get:

\int_1^2 x e^x \, dx = e^2 - e^1

Conclusion

In this article, we demonstrated how to use integration by parts to evaluate the definite integral $\int_1^2 x e^x \, dx$ . We applied the formula for integration by parts and evaluated the resulting terms to obtain the final answer. This technique is a powerful tool for evaluating definite integrals and can be used to solve a wide range of problems in mathematics and physics.

Example Problems

Evaluate the definite integral $\int_0^1 x^2 e^x \, dx$ using integration by parts.
Evaluate the definite integral $\int_1^2 x^3 e^x \, dx$ using integration by parts.
Evaluate the definite integral $\int_0^1 x e^x \, dx$ using integration by parts.

Solutions

To evaluate the definite integral $\int_0^1 x^2 e^x \, dx$ , we can use integration by parts with $f(x) = x^2$ and $g(x) = e^x$ . We have:

\int_0^1 x^2 e^x \, dx = \int_0^1 x^2 e^x \, dx = \left[ x^2 e^x \right]_0^1 - \int_0^1 2x e^x \, dx

Evaluating the first term, we get:

\left[ x^2 e^x \right]_0^1 = 1 e^1 - 0 e^0

Evaluating the second term, we get:

\int_0^1 2x e^x \, dx = \left[ 2x e^x \right]_0^1 - \int_0^1 2 e^x \, dx

Evaluating the first term, we get:

\left[ 2x e^x \right]_0^1 = 2 e^1 - 0 e^0

Evaluating the second term, we get:

\int_0^1 2 e^x \, dx = \left[ 2 e^x \right]_0^1 = 2 e^1 - 2 e^0

Combining the results, we get:

\int_0^1 x^2 e^x \, dx = 1 e^1 - 0 e^0 - (2 e^1 - 0 e^0) + (2 e^1 - 2 e^0)

Simplifying the above equation, we get:

\int_0^1 x^2 e^x \, dx = 1 e^1 - 2 e^1 + 2 e^1 - 2 e^0 + 2 e^0

\int_0^1 x^2 e^x \, dx = 1 e^1

To evaluate the definite integral $\int_1^2 x^3 e^x \, dx$ , we can use integration by parts with $f(x) = x^3$ and $g(x) = e^x$ . We have:

\int_1^2 x^3 e^x \, dx = \int_1^2 x^3 e^x \, dx = \left[ x^3 e^x \right]_1^2 - \int_1^2 3x^2 e^x \, dx

Evaluating the first term, we get:

\left[ x^3 e^x \right]_1^2 = 8 e^2 - 1 e^1

Evaluating the second term, we get:

\int_1^2 3x^2 e^x \, dx = \left[ 3x^2 e^x \right]_1^2 - \int_1^2 6x e^x \, dx

Evaluating the first term, we get:

\left[ 3x^2 e^x \right]_1^2 = 24 e^2 - 3 e^1

Evaluating the second term, we get:

\int_1^2 6x e^x \, dx = \left[ 6x e^x \right]_1^2 - \int_1^2 6 e^x \, dx

Evaluating the first term, we get:

\left[ 6x e^x \right]_1^2 = 48 e^2 - 6 e^1

Evaluating the second term, we get:

\int_1^2 6 e^x \, dx = \left[ 6 e^x \right]_1^2 = 12 e^2 - 6 e^1

Combining the results, we get:

\int_1^2 x^3 e^x \, dx = 8 e^2 - 1 e^1 - (24 e^2 - 3 e^1) + (48 e^2 - 6 e^1) - (12 e^2 - 6 e^1)

Simplifying the above equation, we get:

\int_1^2 x^3 e^x \, dx = 8 e^2 - 1 e^1 - 24 e^2 + 3 e^1 + 48 e^2 - 6 e^1 - 12 e^2 + 6 e^1

\int_1^2 x^3 e^x \, dx = 20 e^2 - 4 e^1

To evaluate the definite integral $\int_0^1 x e^x \, dx$ , we can use integration by parts with $f(x) = x$ and $g(x) = e^x$ . We have:

\int_0^1 x e^x \, dx = \int_0^1 x e^x \, dx = \left[ x e^x \right]_0^1 - \int_0^1 e^x \, dx

Evaluating the first term, we get:

\left[ x e^x \right]_0^1 = 1 e^1 - 0 e^0

Evaluating the second term, we get:

\int_0^1 e^x \, dx = \left[ e^x \right]_0^1 = e^1 - e^0

Combining the results, we get:

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