Grade Select GradeIntegral Calculus

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

Sridhar Ramaswamy

11 Years agoGrade Select Grade

1 Answer

Latika Leekha

11 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.