Practice section 6 of 10

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

Use the given substitutions to find the following integrals:

(a) $\displaystyle\int\dfrac1{e^x+e^{-x}}\,dx$, let $u=e^x$.

(b) $\displaystyle\int\dfrac{x+1}{(2x-1)^2}\,dx$, let $u=2x-1$.

(c) $\displaystyle\int\dfrac1{2+\sqrt x}\,dx$, let $u=\sqrt x$.

(d) $\displaystyle\int\sin^3x\,dx$, let $u=\cos x$.

(e) $\displaystyle\int\dfrac{x}{\sqrt{1-x^2}}\,dx$, let $x=\sin\theta$.

(f) $\displaystyle\int\dfrac1{(x^2+1)^2}\,dx$, let $x=\tan\theta$.

Find the following integrals:

(a) $\displaystyle\int x^2\ln x\,dx$

(b) $\displaystyle\int \ln(x^2+1)\,dx$

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

(d) $\displaystyle\int x^2e^x\,dx$

(e) $\displaystyle\int e^x\cos x\,dx$

(f) $\displaystyle\int \theta\sin\theta\cos\theta\,d\theta$.

(i) Show that $\dfrac d{dx}\sqrt{1-x^2}=\dfrac{-x}{\sqrt{1-x^2}}$, $|x|<1$.

(ii) Hence find $\displaystyle\int\sin^{-1}x\,dx$.