Review practice · Section 1 of 3

(i) Find $\dfrac{d}{dx}(1-e^{4x})^{1/2}$.1 mark

(ii) Hence find $\displaystyle\int e^{2x}\sin^{-1}(e^{2x})\,dx$.4 marks

Find

(i) $\displaystyle\int\dfrac1{1+\cos2x}\,dx$.2 marks

(ii) $\displaystyle\int\dfrac{e^{2x}}{e^x+2}\,dx$, using the substitution $u=e^x+2$.4 marks

(a) Find $\displaystyle\int\dfrac{8x^2+1}{4x^2-1}\,dx$.2 marks

(b) Find, using integration by parts, $\displaystyle\int e^{3x}\tan^{-1}(e^{-3x})\,dx$.4 marks

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

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

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

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

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