Reuse two basic integrals and a radical substitution · source reference

Printed question

(a)(i) Find $\int\frac{2-x}{\sqrt{4x-x^2-1}}dx$. [2]

(ii) Find $\int\frac1{\sqrt{4x-x^2-1}}dx$. [2]

(iii) Using (i) and (ii), find $\int\frac x{\sqrt{4x-x^2-1}}dx$. [2]

(b)(i) By using $u=\sqrt x$, find $\int\frac{\sqrt x}{1+\sqrt x}dx$. [4]

(ii) Find the exact value of $\int_0^9\ln(1+\sqrt x)dx$, leaving the answer $a\ln b-c/b$, where $a,b,c$ are integers to be determined. [4]

[2016/VJC/Promo/11]

Printed working

(a)(i)

$$\int\frac{2-x}{\sqrt{4x-x^2-1}}dx=\frac12\int\frac{4-2x}{\sqrt{4x-x^2-1}}dx=\frac12\frac{(4x-x^2-1)^{1/2}}{1/2}+C=\sqrt{4x-x^2-1}+C.$$

(a)(ii)

$$\int\frac{dx}{\sqrt{4x-x^2-1}}=\int\frac{dx}{\sqrt{3-(x-2)^2}}=\int\frac{1\,dx}{\sqrt{(\sqrt3)^2-(x-2)^2}}=\sin^{-1}\frac{x-2}{\sqrt3}+C.$$

(a)(iii)

$$\int\frac x{\sqrt{4x-x^2-1}}dx=-\int\frac{-x+2-2}{\sqrt{4x-x^2-1}}dx=-\int\frac{2-x}{\sqrt{4x-x^2-1}}dx+\int\frac2{\sqrt{4x-x^2-1}}dx=-\sqrt{4x-x^2-1}+2\sin^{-1}\frac{x-2}{\sqrt3}+C.$$

(b)(i)

$u=\sqrt x$, $du/dx=1/(2\sqrt x)$, $dx/du=2\sqrt x=2u$. Alternatively $u^2=x$, $2u\,du/dx=1$, $dx/du=2u$.

$$\int\frac{\sqrt x}{1+\sqrt x}dx=\int\frac u{1+u}(2u\,du)=\int\frac{2u^2}{1+u}du=\int\left(2u-2+\frac2{1+u}\right)du=u^2-2u+2\ln|1+u|+C=x-2\sqrt x+2\ln(1+\sqrt x)+C,$$

since $1+\sqrt x>0$.

(b)(ii)

$$\begin{aligned}\int_0^9\ln(1+\sqrt x)dx&=[x\ln(1+\sqrt x)]_0^9-\int_0^9x\frac1{1+\sqrt x}\frac1{2\sqrt x}dx\\&=9\ln4-\frac12\int_0^9\frac{\sqrt x}{1+\sqrt x}dx\\&=9\ln4-\frac12[x-2\sqrt x+2\ln(1+\sqrt x)]_0^9\\&=9\ln4-\frac12(9-2\sqrt9+2\ln4)=16\ln2-\frac32.\end{aligned}$$

Teacher notes

(a)(i) power formula; (a)(ii) MF27 $\int dx/\sqrt{a^2-x^2}=\sin^{-1}(x/a)+c$, and the not-in-MF27 composite formula $\int f'(x)/\sqrt{a^2-(f(x))^2}dx=\sin^{-1}(f(x)/a)+c$, where $f=x-2,a=\sqrt3$.

(b)(i) substitution $\int f(x)dx=\int g(u)(dx/du)du$; final answer of an indefinite integral in $x$ must be in $x$. (b)(ii) parts with $u=\ln(1+\sqrt x)$, $dv/dx=1$.

Printed answer key

(a)(i) $\sqrt{4x-x^2-1}+C$; (ii) $\sin^{-1}((x-2)/\sqrt3)+C$; (iii) $-[\sqrt{4x-x^2-1}]+2\sin^{-1}((x-2)/\sqrt3)+C$. (b)(i) $x-2\sqrt x+2\ln|1+\sqrt x|+C$; (ii) $16\ln2-3/2$.