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Integration by Parts Calculator

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Integration by parts is one of the procedure of integration. Integration by parts method will integrate the product of functions and is derived by differentiation the product rule.

Consider the product rule,
$(g(x)h(x))^{'} = g(x)h^{'}(x)+g^{'}(x)h(x)$

Integrating both sides and rearranging,
$(g(x)h(x))^{'} = \int g(x)h^{'}(x) dx+\int g^{'}(x)h(x) dx$
$\int g(x)h^{'}(x)dx = (g(x)h(x))^{'}dx - \int g^{'}(x)h(x) dx$

or

By replacing g(x) and h(x) by u and v,
$\int u \ dv = uv - \int v \ du$

This is the integration by parts formula.

Plug in the function into the integration by parts calculator, this online calculator will integrate the given function.

Steps

Step 1 : Read the given problem and consider first and second function as u and v.

Step 2 : Substitute those values into the integration by parts formula i.e.,
$\int u \ dv = uv - \int v \ du$

and get the integrated function.

Problem

Based on Integration by parts, some of the problems are given below.

Solved Examples

Question 1: Evaluate $\int e^x \ cosx \ dx$.
Solution:

Step 1: Given : $\int e^x \ cosx \ dx$
Let us take,
$u = e^x$ ; $dv = cosx \ dx$
$du = e^x \ dx$ ; $v = sinx$

Step2 : Integrating by parts,
$\int u \ dv = uv - \int v \ du$
$\int e^x \ cosx \ dx = e^x \ sinx - \int e^x \ sinx \ dx$

Again integrating by parts and taking,
$u = e^x$ ; $dv = sinx \ dx$
$du = e^x \ dx$ ; $v = -cosx$

$\int e^x \ cosx \ dx = e^x \ sinx - {e^x \ -cosx - \int e^x \ -cosx \ dx}$
$\int e^x \ cosx \ dx = e^x \ sinx + e^x \ cosx - \int e^x \ cosx \ dx$
$2\int e^x \ cosx \ dx = e^x \ sinx + e^x \ cosx$ + C
$\int e^x \ cosx \ dx = \frac{1}{2}(e^x \ sinx + e^x \ cosx)$ + C

Question 2: Evaluate $\int x^{5}e^{x^3} dx$.
Solution:

Step 1: Given : $\int x^{5}e^{x^3} dx$ = $\int x^{3}x^{2}e^{x^3} dx$
Let us take,
$u = x^3$ ; $dv = x^{2}e^{x^3} dx$
$du = 3x^2 dx$ ; $v = \frac{1}{3}e^{x^3}$

Step2 : Integrating by parts,
$\int u \ dv = uv - \int v \ du$
$\int x^{5}e^{x^3} dx = x^3 \ \frac{1}{3}e^{x^3} - \int \frac{1}{3}e^{x^3} \ (3x^2) dx$
$\int x^{5}e^{x^3} dx = \frac{1}{3}x^{3}e^{x^3} - \int x^{2}e^{x^3}dx$
$\int x^{5}e^{x^3} dx = \frac{1}{3}x^{3}e^{x^3} - \frac{1}{3}e^{x^3}dx$ + C