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Worksheet
Integration Rules (Non-polynomial)
IB Mathematics AA · SL & HL · Shadow Worksheet
Practice
Name
Question 1
Integrate the following:
(a)\(e^{3x}\)
(b)\(2e^{-x}\)
(c)\(\dfrac{1}{x}\)
(d)\(\sin(2x)\)
(e)\(3\cos(4x)\)
(f)\(\sec^2(x)\)
(a)
Answer
\( frac{1}{3}e^{3x}+C\)
(b)
Answer
\(-2e^{-x}+C\)
(c)
Answer
\(\ln|x|+C\)
(d)
Answer
\(- frac{1}{2}\cos(2x)+C\)
(e)
Answer
\( frac{3}{4}\sin(4x)+C\)
(f)
Answer
\( an x+C\)
Question 2
Evaluate:
(a)\(\displaystyle\int_0^1 e^{2x}\,dx\)
(b)\(\displaystyle\int_1^e \dfrac{3}{x}\,dx\)
(c)\(\displaystyle\int_0^{\pi/4}\cos(2x)\,dx\)
(a)
Integrate
\([ frac{1}{2}e^{2x}]_0^1= frac{e^2-1}{2}\)
Answer
\(\dfrac{e^2-1}{2}\)
(b)
Integrate
\([3\ln x]_1^e=3-0\)
Answer
\(3\)
(c)
Integrate
\([ frac{1}{2}\sin(2x)]_0^{\pi/4}= frac{1}{2}\sin( frac{\pi}{2})-0\)
Answer
\(\dfrac{1}{2}\)
Question 3
Find the exact area enclosed between \(y=e^x\), the \(x\)-axis, and the lines \(x=0\) and \(x=\ln 5\).
Integrate
\([e^x]_0^{\ln5}=5-1\)
Answer
\(4\) square units
Question 4
Find \(\displaystyle\int\!\left(2e^x+\dfrac{3}{x}-\cos x ight)dx\).
Answer
\(2e^x+3\ln|x|-\sin x+C\)
Question 5
Find the exact area enclosed between \(y=\sin x\) and the \(x\)-axis for \(0\leq x\leq\pi\).
Integrate
\([-\cos x]_0^\pi=(-\cos\pi)-(-\cos0)=1+1\)
Answer
\(2\) square units
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