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integrate limit pi/4 to 3pi/4 of 1/(1+cosx)

integrate limit pi/4 to 3pi/4 of 1/(1+cosx)

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1 Answers

Latika Leekha
askIITians Faculty 165 Points
9 years ago
Hello student,
we have
\int_{\frac{\pi }{4}}^{\frac{3\pi }{4}}\frac{1}{1+cos x}dx
first consider the term 1/(1+cos x)
It can be written as (1 - cos x)/(1-cos2x) = (1 - cos x)/sin2x
So, we have
\int \frac{1}{sin^{2}x}-\int \frac{cos x}{sin^{2}x}
first term directly gives -cot x
second term is \int \frac{cos x}{sin^{2}x}
For this, let t = sin x
dt = cos x dx
so we have
\int \frac{dt}{t^{2}}
= -1/t
= – cosec x
Hence, the required integral is
\int_{\frac{\pi }{4}}^{\frac{3\pi }{4}}(-cot x + cosec x)dx
= 2.

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