Practice section 7 of 10

(i) State the derivative of $\sin(x^2)$.

(ii) Find $\displaystyle\int x^3\cos(x^2)\,dx$.

You are given that $I=\displaystyle\int x(1-x)^{1/2}\,dx$.

(i) Use integration by parts to find an expression for $I$.

(ii) Use the substitution $u^2=1-x$ to find another expression for $I$.

(iii) Show algebraically that your answers to parts (i) and (ii) differ by a constant.

Given $2\cos A\sin B=\sin(A+B)-\sin(A-B)$, find $\displaystyle\int\cos4\theta\sin2\theta\,d\theta$.

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

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

Find

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

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

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

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