Discussion questions · Section 1 of 2

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

Find the following integrals:

(a) $\displaystyle\int \dfrac1{2-9x^2}\,dx$

(b) $\displaystyle\int \dfrac1{6+4y-y^2}\,dy$

(c) $\displaystyle\int \dfrac1{y^2-4y-13}\,dy$

(d) $\displaystyle\int \dfrac1{2y^2-4y-1}\,dy$

(e) $\displaystyle\int \dfrac1{x^2+4x+20}\,dx$

(f) $\displaystyle\int \dfrac1{2x^2+3x+10}\,dx$

(g) $\displaystyle\int \dfrac1{\sqrt{5-4y-y^2}}\,dy$

(h) $\displaystyle\int \dfrac1{\sqrt{25-9x^2}}\,dx$.

Find the following integrals:

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

(b) $\displaystyle\int \dfrac1{y^2(y-2)}\,dy$

(c) $\displaystyle\int \dfrac1{(x^2+x+1)(x-2)}\,dx$

(d) $\displaystyle\int \dfrac{16x+5}{4x^2+2x+1}\,dx$

(e) $\displaystyle\int \dfrac{4+s}{2s^2+2s+3}\,ds$

(f) $\displaystyle\int \dfrac{2x-1}{x^2+4x+5}\,dx$.

Find the following integrals:

(a) $\displaystyle\int \sec^2(5x)\tan^3(5x)\,dx$

(b) $\displaystyle\int \dfrac{\sin y}{1+2\cos y}\,dy$

(c) $\displaystyle\int e^x\operatorname{cosec}^2(e^x)\,dx$

(d) $\displaystyle\int \tan^2(2t)\,dt$

(e) $\displaystyle\int \dfrac1{1-\cos2x}\,dx$.