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\int \frac{dx}{1+\sin x}

  • Option 1)

    \frac{1}{1+\tan _{\frac{x}{2}} }+C

  • Option 2)

    -\frac{1}{1+\tan _{\frac{x}{2}} }+C

  • Option 3)

    \frac{1}{1+\cot _{\frac{x}{2}} }+C

  • Option 4)

    -\frac{1}{1+\cot _{\frac{x}{2}} }+C

 

Answers (1)

best_answer

As we learned,

 

Type of Integration by perfect square -

 

The integrals are of the from 

(i)    \int \frac{1}{a\cos x+b\sin x}dx

(ii) \int \frac{1}{a+b\cos x}dx

(iii) \int \frac{1}{a+b\sin x}dx

 

- wherein

Working rule :

Resolve :

\cos x=\cos^{2}\frac{x}{2}-\sin^{2}\frac{x}{2}

and 

\sin x=2\sin\frac{x}{2}\cos\frac{x}{2}

 

 

 

\sin x=2\sin\frac{x}{2}\cos\frac{x}{2}

or \sin x=\frac{2\tan _{\frac{x}{2}}}{1+\tan _{\frac{x}{2}}^{2}}

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


Option 1)

\frac{1}{1+\tan _{\frac{x}{2}} }+C

Option 2)

-\frac{1}{1+\tan _{\frac{x}{2}} }+C

Option 3)

\frac{1}{1+\cot _{\frac{x}{2}} }+C

Option 4)

-\frac{1}{1+\cot _{\frac{x}{2}} }+C

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gaurav

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