Discussion questions · Section 2 of 2

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 $\displaystyle\int\sin^{-1}x\,dx$.

(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 $\displaystyle\int\cos4\theta\sin2\theta\,d\theta$.