\;{dx}{≤ft(x-π/3) x}=

Correct Answer is : 2log⁡( tan⁡x−32)+C2≤ft(\,{ x-{3}}{2})+C2log(2tanx−3)+C

Correct Answer is : 2log⁡( tan⁡x−32)+C2≤ft(\,{ x-{3}}{2})+C2log(2tanx−3)+C

Test Index

AP EAPCET 08-Jul-2022 Shift 1 Paper

Section: Mathematics

Question : 72 of 160

Marks: +1, -0

\int\;\frac{dx}{\sin\left(x-\frac{\pi}{3}\right)\cos x}=

Solution: 👈: Video Solution

Let

I=\int\frac{1}{\sin\left(x-\frac{\pi}{3}\right)\cos x}\,dx

I=2\int\frac{\cos\frac{\pi}{3}}{\sin\left(x-\frac{\pi}{3}\right)\cos x}\,dx=2\int\frac{\cos\left(x-\frac{\pi}{3}-x\right)}{\sin\left(x-\frac{\pi}{3}\right)\cos x}\,dx

=2\int\left[\frac{\cos\left(x-\frac{\pi}{3}\right)\cos x}{\sin\left(x-\frac{\pi}{3}\right)\cos x}+\frac{\sin\left(x-\frac{\pi}{3}\right)\sin x}{\sin\left(x-\frac{\pi}{3}\right)\cos x}\right]\,dx

=2\int\left[\cot\left(x-\frac{\pi}{3}\right)+\tan x\right]\,dx

=2\left[\log\left|\sin\left(x-\frac{\pi}{3}\right)\right|-\log\left|\cos x\right|\right]+C

=2\left[\log\left|\frac{\sin x\cos\frac{\pi}{3}}{\cos x}-\frac{\cos x\sin\frac{\pi}{3}}{\cos x}\right|\right]+C

=2\left[\log\left|\frac{\tan x-\sqrt{3}}{2}\right|\right]+C

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