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Integrate \int \frac{dx}{\sin ^{2}x+2\sin x\cos x}

  • Option 1)

    \ln \frac{t+1}{t-1}+c

  • Option 2)

    \ln \frac{t+2}{t}+c

  • Option 3)

    \ln \frac{t}{t+2}+c

  • Option 4)

    none of these

 

Answers (1)

best_answer

As we learned,

 

Type of Integration by perfect square -

Integration are in the format 

(i) \int \frac{dx}{a^{2}\sin ^{2}x+2b\sin x\cos x+c^{2}\cos^{2}x}

(ii) \int \frac{dx}{a\cos^{2}x+b} (iii) \int \frac{dx}{a+b\sin^{2}x}

- wherein

Working rule :

Divide by cos^{2}x in each case and pull t=\tan x ,  dt=\sec^{2}xdx

 

 

Divide by cos^{2}x

\int \frac{dx\sec ^{2}x}{\tan ^{2}x+2\tan x}=\int \frac{dt}{\left ( t+1 \right )^{2}-1}=\ln \frac{t}{\left ( t+2 \right )}+C


Option 1)

\ln \frac{t+1}{t-1}+c

Option 2)

\ln \frac{t+2}{t}+c

Option 3)

\ln \frac{t}{t+2}+c

Option 4)

none of these

Posted by

gaurav

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