Integration by Parts: Problems with Solutions

By Hernando Guzman Jaimes (prof. at University of Zulia - Maracaibo, Venezuela)
Problem 1
Evalutate the integral $\int x^{3}\ln\ x\ dx$, using integration by parts.
Let
$u=\ln x,$
$dv=x^{3}dx.$
Problem 2
Let $u=4x+7,$ $dv=e^{x}dx$

Evaluate the integral $\int (4x+7)e^{x}dx$ using integration by parts.
Problem 3
Evalutate the integral $\int x\sin 3x$ $dx$ using intregation by parts.
Consider $u=x,$
$dv=\sin 3x$ $dx$.
Problem 4
Using integration by parts find the integral $\int x\cos 4x\ dx$,
Consider $ u=x,$
$dv=\cos 4x$ $dx.$
Problem 5
Using integration by parts find the integral
$\int \cos ^{3}x\sin x\ dx$.
Problem 6
Using integration by parts find the integral:
$\int \cos ^{3}x\sin ^{4}x\ dx$
Problem 7
Evalutate the following integral
$\int \sin ^{3}x\cos ^{2}x\ dx$
Problem 8
$\int \sin^{3}x\ dx=$
Problem 9
Evaluate the following integral
$\int xe^{-2x}dx$
using the method of integration by parts.
Problem 10
$\int \frac{2x}{e^{x}}\ dx =$

Problem 11
$\int x^{2}e^{x}\ dx = $
Problem 12
$\int t$ $\ln (t+1)$ $dt=$
Problem 13
$\int \frac{\ln (x)}{x^{2}}dx= $
Problem 14
Evaluate the integral using integration by parts.
$\int x\cos x$ $dx$
Problem 15
Evaluate the integral $\int x\sin x$ $dx$ using the method of integration by parts.
Problem 16
$\int x^{2}\sin x$ $dx=$
Problem 17
Determine the solution of the integral
$\int e^{2x}\sin x\ dx$
Problem 18
$I=\int e^{-3x}\sin 5x\ dx = $
Problem 19
Determine the solution of the following integral
$\int \sec ^{2}x\tan x\ dx$
Problem 20
Solve the following trigonometric integral
$\int \tan ^{2}x\sec ^{4}x\ dx$
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