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Question: \(\int_{}^{}\frac{dx}{a^{2}\cos^{2}x + b^{2}\sin^{2}x}\)is equal to...

∫dxa2cos⁡2x+b2sin⁡2x\int_{}^{}\frac{dx}{a^{2}\cos^{2}x + b^{2}\sin^{2}x}is equal to

A

tan⁡−1(abtan⁡x)+C\tan^{- 1}\left( \frac{a}{b}\tan x \right) + C

B

1abtan⁡−1(bacot⁡x)+C\frac{1}{ab}\tan^{- 1}\left( \frac{b}{a}\cot x \right) + C

C

1abtan⁡−1(batan⁡x)+C\frac{1}{ab}\tan^{- 1}\left( \frac{b}{a}\tan x \right) + C

D

tan⁡−1(batan⁡x)+C\tan^{- 1}\left( \frac{b}{a}\tan x \right) + C

Answer

1abtan⁡−1(batan⁡x)+C\frac{1}{ab}\tan^{- 1}\left( \frac{b}{a}\tan x \right) + C

Explanation

Solution

Putting t=tan⁡xt = \tan x, we get

∫dxa2cos⁡2x+b2sin⁡2x=∫sec⁡2xdxa2+b2tan⁡2x=∫dta2+b2t2\int \frac { d x } { a ^ { 2 } \cos ^ { 2 } x + b ^ { 2 } \sin ^ { 2 } x } = \int \frac { \sec ^ { 2 } x d x } { a ^ { 2 } + b ^ { 2 } \tan ^ { 2 } x } = \int \frac { d t } { a ^ { 2 } + b ^ { 2 } t ^ { 2 } }

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