Definite Integration Question 53
Question 53
- The value of the integral $\int_{0}^{1 / 2} \frac{1+\sqrt{3}}{\left((x+1)^{2}(1-x)^{6}\right)^{1 / 4}} d x$ is
(2018 Adv.)
Fill in the Blanks
Show Answer
Solution:
- (2) Let $I=\int_{0}^{1 / 2} \frac{1+\sqrt{3}}{\left[(x+1)^{2}(1-x)^{6}\right]^{1 / 4}} d x$
$$ \Rightarrow \quad I=\int_{0}^{1 / 2} \frac{1+\sqrt{3}}{(1-x)^{2} \frac{1-x}{1+x}} d x $$
Put $\quad \frac{1-x}{1+x}=t \Rightarrow \frac{-2 d x}{(1+x)^{2}}=d t$
when $x=0, t=1, x=\frac{1}{2}, t=\frac{1}{3}$
$\therefore$
$I=\int_{1}^{1 / 3} \frac{(1+\sqrt{3}) d t}{-2(t)^{6 / 4}}$
$\Rightarrow \quad I=\frac{-(1+\sqrt{3})}{2} \frac{-2}{\sqrt{t}}_{1}^{1 / 3}$
$\Rightarrow \quad I=(1+\sqrt{3})(\sqrt{3}-1) \Rightarrow I=3-1=2$
