Integration by partsAQA A-Level Maths: Mind map

Exam questions on Integration by parts

1. Let $I=\int x\,e^{2x}\,dx$.
   Hence find the exact value of $\int_0^1x\,e^{2x}\,dx$.2 marks
2. A student tries to find $\int x\cos x\,dx$ by integration by parts, choosing $u=\cos x$ and $\frac{dv}{dx}=x$.
   Hence find the exact value of $\int_0^{\pi/2}x\cos x\,dx$.2 marks
3. The function $y=\ln x$ is defined for $x>0$.
   Use integration by parts with $u=\ln x$ to show that $\int\ln x\,dx=x\ln x-x+c$.3 marks