Review practice · Section 2 of 3

Find the following integrals:

(a) $\displaystyle\int\dfrac1{1+\cos4\theta}\,d\theta$,2 marks

(b) $\displaystyle\int\cos2x\sin3x\,dx$, given $2\sin A\cos B=\sin(A+B)+\sin(A-B)$.3 marks

(a) By using the substitution $u=\sqrt x$, find $\displaystyle\int\dfrac1{\sqrt x\,(x+2\sqrt x+2)}\,dx$.4 marks

(b) Find $\dfrac d{dx}(x\sin x)$.1 mark

Hence find $\displaystyle\int(\sin x+x\cos x)\ln x\,dx$.3 marks

Use the substitution $u=\cos2\theta$ to find $\displaystyle\int\dfrac{\sin2\theta}{4-\cos^22\theta}\,d\theta$.3 marks

(i) Find $\displaystyle\int\dfrac{e^{\tan^{-1}2x}}{1+4x^2}\,dx$.2 marks

(ii) Given that $-\dfrac\pi2<\theta<\dfrac\pi2$, use the substitution $x=\sin\theta$ to find $\displaystyle\int\dfrac{x^2}{\sqrt{1-x^2}}\,dx$.6 marks

Find the following integrals:

(a) $\displaystyle\int\dfrac{e^{2x}}{e^{2x}-3}\,dx$2 marks

(b) $\displaystyle\int\dfrac{3x^2+2}{\sqrt{x^3+2x-8}}\,dx$2 marks

(c) $\displaystyle\int\dfrac1{x^2+2x+5}\,dx$2 marks

(d) $\displaystyle\int x^2\ln x\,dx$3 marks

(e) $\displaystyle\int x\sec(x^2)\tan(x^2)\,dx$2 marks

Express $x+3=A(2x+2)+B$, where $A$ and $B$ are constants to be determined.

Hence, find $\displaystyle\int\dfrac{x+3}{x^2+2x+2}\,dx$.3 marks