Question

If $\int \frac{1-(\cot x)^{2019}}{\tan x+(\cot x)^{2020}} d x=\frac{1}{n} \ln \left|(f(x))^{n}+(g(x))^{n}\right|+c$, then the value of $n\left[(f(x))^{4}+(g(x))^{4}\right]_{x=\frac{\pi}{3}}=$

• A. $\frac{10105}{16}$
• B. $\frac{10012}{15}$
• C. $\frac{20210}{9}$
• D. $\frac{10105}{8}$

Answer 1

Answer: (D)

Let $I=\int \frac{1-(\cot x)^{2019}}{\tan x+(\cot x)^{2020}} d x$
$=\int \frac{1-\frac{(\cos x)^{2019}}{(\sin x)^{2019}}}{\frac{\sin x}{\cos x}+\frac{(\cos x)^{2020}}{(\sin x)^{2020}} d x}$
$=\int \frac{\sin ^{2019} x-\cos ^{2019} x}{\sin ^{2021} x+\cos ^{2021} x} \cdot \sin x \cos x d x$
Put
$\sin ^{2021} x+\cos ^{2021} x=t$
$\Rightarrow \left[2021 \sin ^{2020} x \cdot \cos x+2021 \cos ^{2020} \cdot(-\sin x)\right]$
$d x=d t$
$\Rightarrow \sin x \cos x\left[\sin ^{2019} x-\cos ^{2019} x\right] d x=\frac{1}{2021} d t$
$\therefore I=\frac{1}{2021} \int \frac{1}{t} d t=\frac{1}{2021} \log t+C$
$=\frac{1}{2021} \log \left[\sin ^{2021} x+\cos ^{2021} x\right]+C$
$\therefore f(x)=\sin x, g(x)=\cos x, n=2021$
$\therefore n\left[\left(f(x)^{4}+\left(g(x)^{4}\right)\right]_{x=\frac{\pi}{3}}=2021\left[\sin ^{4} \frac{\pi}{3}+\cos ^{4} \frac{\pi}{3}\right]\right.$
$=2021\left[\left(\frac{\sqrt{3}}{2}\right)^{4}+\left(\frac{1}{2}\right)^{4}\right]$
$=2021\left[\frac{9}{16}+\frac{1}{16}\right]=\frac{2021 \times 10}{16}=\frac{10105}{8}$