Find \( \displaystyle\int e^{2x}\, dx \).
\( \dfrac{1}{2}e^{2x}+C \).
Further Integration
10 exam-style questions on integration by parts, substitution, partial fractions and differential equations. Tap Show answer for full working.
3 Easy4 Medium3 Hard
Find \( \displaystyle\int \sin 3x\, dx \).
\( -\dfrac{1}{3}\cos 3x + C \).
Evaluate \( \displaystyle\int_0^1 \dfrac{1}{1+x}\, dx \).
\( [\ln|1+x|]_0^1=\ln 2 \).
Use integration by parts to find \( \displaystyle\int x e^x\, dx \).
\( xe^x - \int e^x\, dx = xe^x - e^x + C = e^x(x-1)+C \).
Use the substitution \( u=1+x^2 \) to find \( \displaystyle\int \dfrac{2x}{1+x^2}\, dx \).
\( \int \dfrac{1}{u}\, du = \ln|1+x^2| + C \).
Express \( \dfrac{3}{(x-1)(x+2)} \) in partial fractions and hence integrate.
\( \dfrac{1}{x-1}-\dfrac{1}{x+2} \). Integral \( \ln\left|\dfrac{x-1}{x+2}\right|+C \).
Solve the differential equation \( \dfrac{dy}{dx}=3y \) given that \( y=2 \) when \( x=0 \).
\( \dfrac{1}{y}dy=3\,dx \) ⇒ \( \ln|y|=3x+C \). \( y=Ae^{3x} \). \( A=2 \). \( y=2e^{3x} \).
Find \( \displaystyle\int x\cos x\, dx \).
Parts: \( x\sin x - \int\sin x\, dx = x\sin x + \cos x + C \).
Evaluate \( \displaystyle\int_0^{\pi/2} x\sin x\, dx \).
\( [-x\cos x]_0^{\pi/2} + \int_0^{\pi/2}\cos x\, dx = 0 + [ \sin x ]_0^{\pi/2}=1 \).
Solve \( \dfrac{dy}{dx}=\dfrac{y}{x+1} \) given \( y=4 \) when \( x=0 \).
\( \dfrac{1}{y}dy=\dfrac{1}{x+1}dx \) ⇒ \( \ln|y|=\ln|x+1|+C \) ⇒ \( y=A(x+1) \). \( A=4 \). \( y=4(x+1) \).