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evaluate the following integral ∫ 1 / sqrt(1 + cos x) dx

evaluate the following integral
∫ 1 / sqrt(1 + cos x) dx

Grade:12

1 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
6 years ago
Ans:
Hello Student,
Please find answer to your question below

I = \int \frac{1}{\sqrt{1+cosx}}dx
I = \int \frac{1}{\sqrt{2cos^{2}\frac{x}{2}}}dx
I = \int \frac{1}{\sqrt{2}cos\frac{x}{2}}dx
I = \int \frac{sec\frac{x}{2}}{\sqrt{2}}dx
I = \int \frac{sec\frac{x}{2}(sec\frac{x}{2}+tan\frac{x}{2})}{\sqrt{2}(sec\frac{x}{2}+cot\frac{x}{2})}dx
sec\frac{x}{2}+tan\frac{x}{2} = t
\frac{1}{2}(sec\frac{x}{2}tan\frac{x}{2}+sec^{2}\frac{x}{2})dx = dt
I = \int \frac{2}{\sqrt{2}t}dt
I = \sqrt{2}logt+c
I = \sqrt{2}log(sec\frac{x}{2}+tan\frac{x}{2})+c

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