Practice section 5 of 10

Integrate with respect to $x$: (a) $(2x+3)\ln x$; (b) $x\sin x$; (c) $\ln x$; (d) $x^2\cos x$; (e) $e^{2x}\sin x$.

Integrate with respect to $x$: (a) $x\ln x$; (b) $x\cos x$; (c) $xe^{-x}$.

(i) Find the derivative of $e^{x^2}$. (ii) Hence find $\int x^3e^{x^2}\,dx$.

Find $\dfrac{d}{dx}\left(\dfrac{x^3}{1+x^4}\right)$ and hence $\displaystyle\int\dfrac{4x^2(3-x^4)}{(1+x^4)^2}\,dx$.

Show that $\dfrac d{dx}(\tan^3x)=3\sec^4x-3\sec^2x$. Hence find $\displaystyle\int\sec^4x\,dx$.

Find the following integrals:

(a) $\displaystyle\int \sqrt{x}(x+3)\,dx$

(b) $\displaystyle\int \left(\sqrt y+\dfrac1{\sqrt y}\right)^2\,dy$

(c) $\displaystyle\int \dfrac{x+1}{(x^2+2x+3)^3}\,dx$

(d) $\displaystyle\int \dfrac{\ln x}x\,dx$

(e) $\displaystyle\int \dfrac{3t}{1-t}\,dt$

(f) $\displaystyle\int \dfrac1{x\ln x}\,dx$

(g) $\displaystyle\int \dfrac{e^x}{1+e^x}\,dx$

(h) $\displaystyle\int \dfrac{e^{\sqrt x}}{\sqrt x}\,dx$

(i) $\displaystyle\int \dfrac{e^{1/x}}{x^2}\,dx$

(j) $\displaystyle\int 2017^{2017x}\,dx$.