Mathski
Worksheet
Integration by Parts
IB Mathematics AA · HL · Shadow Worksheet
Practice
Name
Question 1
Use integration by parts to find:
(a)\(\displaystyle\int xe^x\,dx\)
(b)\(\displaystyle\int x\sin x\,dx\)
(c)\(\displaystyle\int x\ln x\,dx\)
(a)
\(u=x,\;dv=e^x\,dx\Rightarrow du=dx,\;v=e^x\)
\(xe^x-\displaystyle\int e^x\,dx\)
Answer
\(xe^x-e^x+C=e^x(x-1)+C\)
(b)
\(u=x,\;dv=\sin x\,dx\Rightarrow du=dx,\;v=-\cos x\)
\(-x\cos x+\displaystyle\int\cos x\,dx\)
Answer
\(-x\cos x+\sin x+C\)
(c)
\(u=\ln x,\;dv=x\,dx\Rightarrow du= frac{1}{x}\,dx,\;v= frac{x^2}{2}\)
\( frac{x^2}{2}\ln x-\displaystyle\int frac{x}{2}\,dx\)
Answer
\(\dfrac{x^2}{2}\ln x-\dfrac{x^2}{4}+C\)
Question 2
Find \(\displaystyle\int x^2e^x\,dx\) using integration by parts twice.
First application: \(u=x^2,\;dv=e^x\,dx\)
\(x^2e^x-2\displaystyle\int xe^x\,dx\)
Second application (from Q1a)
\(\displaystyle\int xe^x\,dx=e^x(x-1)\)
Answer
\(e^x(x^2-2x+2)+C\)
Question 3
Evaluate \(\displaystyle\int_1^e x^2\ln x\,dx\).
\(u=\ln x,\;dv=x^2\,dx\Rightarrow v= frac{x^3}{3}\)
\([ frac{x^3}{3}\ln x]_1^e-\displaystyle\int_1^e frac{x^2}{3}\,dx\)
Evaluate
\( frac{e^3}{3}-0-[ frac{x^3}{9}]_1^e= frac{e^3}{3}- frac{e^3}{9}+ frac{1}{9}= frac{2e^3}{9}+ frac{1}{9}\)
Answer
\(\dfrac{2e^3+1}{9}\)
Question 4
Find \(\displaystyle\int e^x\sin x\,dx\) using the cyclic method.
Apply twice
Let \(I=\displaystyle\int e^x\sin x\,dx\). First: \(u=\sin x\Rightarrow I=e^x\sin x-\displaystyle\int e^x\cos x\,dx\). Second: \(u=\cos x\Rightarrow I=e^x\sin x-e^x\cos x-I\)
Solve for \(I\)
\(2I=e^x(\sin x-\cos x)\)
Answer
\(\dfrac{e^x(\sin x-\cos x)}{2}+C\)
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