Practice section 8 of 10

Use the substitution $y=1-2x$ to find $\displaystyle\int\dfrac{x^2+1}{\sqrt{1-2x}}\,dx$. [4]

(i) Show that $\displaystyle\int\dfrac{x}{\sqrt{1-x^2}}\,dx=-\sqrt{1-x^2}+c$, where $c$ is an arbitrary constant. [2]

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

(a) Use the substitution $u=\sqrt{x+1}$ to find $\displaystyle\int\dfrac{x^2}{\sqrt{x+1}}\,dx$. [4]

(b) Find $\displaystyle\int\dfrac{e^{5x}}{(e^{5x}-e)^4}\,dx$. [3]

Find the following integrals:

(a) $\displaystyle\int\dfrac1{1+\cos4\theta}\,d\theta$, [2]

(b) $\displaystyle\int\cos2x\sin3x\,dx$, given $2\sin A\cos B=\sin(A+B)+\sin(A-B)$. [3]

(a) By using the substitution $u=\sqrt x$, find $\displaystyle\int\dfrac1{\sqrt x\,(x+2\sqrt x+2)}\,dx$. [4]

(b) Find $\dfrac d{dx}(x\sin x)$. [1]

Hence find $\displaystyle\int(\sin x+x\cos x)\ln x\,dx$. [3]

Use the substitution $u=\cos2\theta$ to find $\displaystyle\int\dfrac{\sin2\theta}{4-\cos^22\theta}\,d\theta$. [3]