Match a denominator derivative and remove a radical · source reference

Printed question

(a) Given that $A(\sin\theta+\cos\theta)+B(\cos\theta-\sin\theta)\equiv4\sin\theta$, find the constants $A,B$. [2]

Hence find the exact value of $\int_0^{\pi/4}\frac{4\sin\theta}{\sin\theta+\cos\theta}d\theta$. [3]

(b)(i) Express $\frac{2t}{(t+1)^2}$ in partial fractions. [2]

(ii) Hence find the exact value of $\int_1^5\frac1{x+\sqrt{2x-1}}dx$, using $t=\sqrt{2x-1}$. [6]

[2010/SAJC/Prelim/I/11]

Printed working

(a)

Compare coefficients of $\sin\theta$ and $\cos\theta$: $A-B=4$ (1), $A+B=0$ (2). Solving gives $A=2$, $B=-2$.

$$\begin{aligned}\int_0^{\pi/4}\frac{4\sin\theta}{\sin\theta+\cos\theta}d\theta&=\int_0^{\pi/4}\frac{2(\sin\theta+\cos\theta)-2(\cos\theta-\sin\theta)}{\sin\theta+\cos\theta}d\theta\\&=\int_0^{\pi/4}2d\theta-2\int_0^{\pi/4}\frac{\cos\theta-\sin\theta}{\sin\theta+\cos\theta}d\theta\\&=2[\theta]_0^{\pi/4}-2[\ln(\sin\theta+\cos\theta)]_0^{\pi/4}\\&=\frac\pi2-\ln2\quad\text{(or }\frac\pi2-2\ln\sqrt2\text{)}.\end{aligned}$$

(b)(i)

$$\frac{2t}{(t+1)^2}=\frac A{t+1}+\frac B{(t+1)^2},\qquad A(t+1)+B=2t.$$

Substitute $t=-1$: $B=-2$; $t=0$: $A=2$.

$$\frac{2t}{(t+1)^2}=\frac2{t+1}-\frac2{(t+1)^2}.$$

(b)(ii)

$t=\sqrt{2x-1}$ gives $dt/dx=2/(2\sqrt{2x-1})$, $dx/dt=\sqrt{2x-1}=t$. Alternatively $t^2=2x-1$, $2t=2dx/dt$. When $x=1,t=1$; when $x=5,t=3$.

$$\begin{aligned}\int_1^5\frac{dx}{x+\sqrt{2x-1}}&=\int_1^3\frac t{(t^2+1)/2+t}dt=\int_1^3\frac{2t}{t^2+2t+1}dt\\&=\int_1^3\frac{2t}{(t+1)^2}dt=2\int_1^3\left(\frac1{t+1}-\frac1{(t+1)^2}\right)dt\\&=2[\ln(t+1)+1/(t+1)]_1^3=2\ln2-\frac12.\end{aligned}$$

Teacher notes

(a) Use $\int f'(\theta)/f(\theta)d\theta=\ln|f(\theta)|+c$.

(b) Let $x=g(t)$ and $dx/dt=g'(t)$. Then $\int f(x)dx=\int f(g(t))(dx/dt)dt=\int f(g(t))g'(t)dt$. Express the integrand in $t$. For a definite integral remember to change the limits from $x=\cdots$ to $t=\cdots$: $\int_{x=1}^{x=5}f(x)dx=\int_{t=1}^{t=3}F(t)dt$.

Use $\int f'(t)/f(t)dt=\ln|f(t)|+c$ and the power formula, with $f(t)=t+1$, $n=-2$.

Printed answer key

(a) $A=2,B=-2$; $\pi/2-\ln2$. (b)(i) $2t/(t+1)^2=2/(t+1)-2/(t+1)^2$; (ii) $2\ln2-1/2$.