Practice section 9 of 10

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

(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]

Find the following integrals:

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

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

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

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

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

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]

Find $\displaystyle\int\dfrac{x+2}{\sqrt{1-8x-4x^2}}\,dx$. [4]

(i) Express $\dfrac{4+x}{(1-x)(4+x^2)}$ in partial fractions. [2]

(ii) Hence find $\displaystyle\int\dfrac{4+x}{(1-x)(4+x^2)}\,dx$. [2]

Find $\displaystyle\int e^{2x}\cos4x\,dx$. [4]