Mathski
Worksheet
Integration by Substitution
IB Mathematics AA · HL · Shadow Worksheet
Practice
Name
Question 1
Use the substitution given to integrate:
(a)\(\displaystyle\int 2x(x^2+1)^4\,dx\), \(u=x^2+1\)
(b)\(\displaystyle\int \dfrac{3x^2}{x^3+5}\,dx\), \(u=x^3+5\)
(c)\(\displaystyle\int x\sqrt{x^2-3}\,dx\), \(u=x^2-3\)
(a)
\(du=2x\,dx\)
\(\displaystyle\int u^4\,du= frac{u^5}{5}\)
Answer
\(\dfrac{(x^2+1)^5}{5}+C\)
(b)
\(du=3x^2\,dx\)
\(\displaystyle\int frac{1}{u}\,du=\ln|u|\)
Answer
\(\ln|x^3+5|+C\)
(c)
\(du=2x\,dx\Rightarrow x\,dx= frac{du}{2}\)
\(\displaystyle\int frac{1}{2}\sqrt{u}\,du= frac{1}{3}u^{3/2}\)
Answer
\(\dfrac{1}{3}(x^2-3)^{3/2}+C\)
Question 2
Choose a suitable substitution and integrate:
(a)\(\displaystyle\int\sin^3x\cos x\,dx\)
(b)\(\displaystyle\int\dfrac{e^x}{e^x+3}\,dx\)
(c)\(\displaystyle\int x^2e^{x^3}\,dx\)
(a)
\(u=\sin x,\;du=\cos x\,dx\)
\(\displaystyle\int u^3\,du= frac{u^4}{4}\)
Answer
\(\dfrac{\sin^4 x}{4}+C\)
(b)
\(u=e^x+3,\;du=e^x\,dx\)
\(\displaystyle\int frac{1}{u}\,du\)
Answer
\(\ln(e^x+3)+C\)
(c)
\(u=x^3,\;du=3x^2\,dx\)
\(\displaystyle\int frac{1}{3}e^u\,du\)
Answer
\(\dfrac{1}{3}e^{x^3}+C\)
Question 3
Evaluate \(\displaystyle\int_0^1 xe^{x^2}\,dx\) using the substitution \(u=x^2\).
Change limits
\(x=0\Rightarrow u=0;\;x=1\Rightarrow u=1\). \(du=2x\,dx\Rightarrow x\,dx= frac{du}{2}\)
Integral becomes
\(\displaystyle\int_0^1 frac{1}{2}e^u\,du=[ frac{1}{2}e^u]_0^1\)
Answer
\(\dfrac{e-1}{2}\)
Question 4
Use the substitution \(u=\ln x\) to find \(\displaystyle\int\dfrac{(\ln x)^2}{x}\,dx\).
\(du= frac{1}{x}\,dx\)
\(\displaystyle\int u^2\,du= frac{u^3}{3}\)
Answer
\(\dfrac{(\ln x)^3}{3}+C\)
Generated by Mathski  ·  mathski.io  ·  IB Mathematics AA Shadow Worksheets