5.16 Integration by substitution and by parts questions for IB Maths AA

Use the substitution $x=3\sin \theta$ to show that $A(a)={\displaystyle \frac{9}{2}}\arcsin \left({\displaystyle \frac{a}{3}}\right)+{\displaystyle \frac{a}{2}}\sqrt{9-a^2}$.

Find the exact value of $A(3)$.

Find the value of $a$ for which $A(a)=5$.

Use your graphic display calculator to find the value of $I_1$, the value of $I_2$ and the value of $I_3$.

Suggest an exact value for $I_n$.

Show that $\sin ({\displaystyle \frac{\pi }{2}}-x)=\cos x$ and $\cos ({\displaystyle \frac{\pi }{2}}-x)=\sin x$.

By using the substitution $u={\displaystyle \frac{\pi }{2}}-x$, show that

Hence prove that your suggestion in part (b) is correct for all $n\in {\mathbb{Z}}^{+}$.

Now consider a function $f$ that is continuous for $0\le x\le a$, where $a>0$, and such that $f(x)+f(a-x)\mathrm{\ne }0$ for $0\le x\le a$.

Show that

Hence find the value of ${\displaystyle {\int }_0^4\frac{\sqrt{x}}{\sqrt{x}+\sqrt{4-x}}dx}$.

Now let $g$ be an even function that is continuous for $-a\le x\le a$.

Show that

Hence find the exact value of ${\displaystyle {\int }_{-\frac{\pi }{3}}^{\frac{\pi }{3}}\frac{\cos x}{1+e^x}dx}$.

Show that $g(\pi -x)=g(x)$.

Let $I={\displaystyle {\int }_0^{\pi }xg(x)dx}$. By using the substitution $u=\pi -x$, show that

By using the substitution $v=\cos x$, show that ${\displaystyle {\int }_0^{\pi }g(x)dx={\int }_{-1}^1\frac{1}{1+v^2}dv}$.

Hence find the exact value of ${\displaystyle {\int }_0^{\pi }g(x)dx}$.

Hence find the exact value of ${\displaystyle {\int }_0^{\pi }\frac{x\sin x}{1+{\cos }^{2}x}dx}$.

By using a similar method, or otherwise, find the exact value of ${\displaystyle {\int }_0^{\pi }x{\sin }^{3}xdx}$.

Use the substitution $u=\sqrt{x}$ to show that ${\displaystyle \int e^{\sqrt{x}}\mathrm{d}x=\int 2u{e}^u\mathrm{d}u}$.

Hence find the exact value of ${\displaystyle {\int }_0^4{e}^{\sqrt{x}}\mathrm{d}x}$.

Find the value of $k$ such that ${\displaystyle {\int }_0^k{e}^{\sqrt{x}}\mathrm{d}x=30}$.

Show that ${\displaystyle {\int }_0^1{f}_n(x)\mathrm{d}x=1}$ for all $n\in {\mathbb{Z}}^{+}$.

Consider the case $n=3$.

Find $\mathrm{E}(X_3)$.

Find the median of $X_3$, and hence state whether the median of $X_3$ is greater than or less than $\mathrm{E}(X_3)$.

By using the substitution $u=1-x$, show that $\mathrm{E}(X_n)={\displaystyle \frac{1}{n+1}}$.

Show that $\mathrm{V}\mathrm{a}\mathrm{r}(X_n)={\displaystyle \frac{n}{(n+1{)}^2(n+2)}}$.

Show that the median of $X_n$ is $m_n=1-2^{-\frac{1}{n}}$.

Show that, for $n\in {\mathbb{Z}}^{+}$, the inequality $\mathrm{E}(X_n)>m_n$ is equivalent to ${(1+{\displaystyle \frac{1}{n}})}^n>2$.

By using the binomial theorem, prove that $\mathrm{E}(X_n)>m_n$ for all integers $n\ge 2$.

Use l'Hôpital's rule to find ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }n{m}_n}$.

Hence find ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{m_n}{\mathrm{E}(X_n)}}$.