Notes: Compute the integral of sin x/(1+cos x)^0.5.

Wednesday, April 30, 2014

We'll re-write the denominator of the
fraction:

(1+cos x)^0.5 = sqrt(1+cos
x)

You'll have to substitute the expression within brackets
by another variable.

Let 1 + cos x =
t

We'll differentiate both
sides:

-sin x dx = dt => sin x dx =
-dt

We'll evaluate the
integral:

$\displaystyle\int$ sin xdx/sqrt(1+cos x) = $\displaystyle\int$ -dt/sqrt
t

$\displaystyle\int$ -t^(-1/2)*dt = - t^(-1/2 + 1)/(-1/2 + 1) +
C

$\displaystyle\int$ -t^(-1/2)*dt = - t^(1/2)/(1/2) +
C

$\displaystyle\int$ -t^(-1/2)*dt = - 2 sqrt t +
C

$\displaystyle\int$ sin xdx/sqrt(1+cos x) = -2 sqrt (1 +
cos x) + C

Notes