Practice section 3 of 10

Prove that $\int\frac1{a^2-x^2}\,dx=\frac1{2a}\ln\left|\frac{a+x}{a-x}\right|+C$.

Express $\frac{x^2+x}{(x-1)(x^2+1)}$ in the form $\frac A{x-1}+\frac{Bx+C}{x^2+1}$. Hence find $\int\frac{x^2+x}{(x-1)(x^2+1)}\,dx$.

Find $\int\frac{6x+16}{x^2+4x+13}\,dx$.

Find $\int\frac{x+1}{9-x^2}\,dx$.

Find (a) $\int\frac1{(1+x)(2-x)}\,dx$; (b) $\int\frac{x}{(x+1)^2}\,dx$.

Integrate with respect to $x$: (a) $\sin(2-3x)$; (b) $\sec(3x)\tan(3x)$; (c) $x\cos(3x^2)$.