1. ## Integrals

Can someone plz help me evaluate these integrals?

integral(
e^x(3 + e^x)^8 )dx

integral(PI/6 PI/4) (sin(x)/(1+cos(x))^2)dx

2. In both cases you can find a primitive quite quickly. Observe that

$\displaystyle \frac{d}{dx}(3+e^x)=e^x$,

so the chain rule yields, for a well-behaved function $\displaystyle g$:

$\displaystyle \frac{d}{dx}g(3+e^x)=e^xg'(3+e^x)$.

In particular, choosing $\displaystyle g(x)=x^9$:

$\displaystyle \frac{d}{dx}(3+e^x)^9=9e^x(3+e^x)^8$.

Consequently

$\displaystyle \int e^x(3+e^x)^8\; dx=\frac{1}{9}(3+e^x)^9$.

Try applying the same method to the second one

3. Originally Posted by Sally_Math
Can someone plz help me evaluate these integrals?

integral(
e^x(3 + e^x)^8 )dx

integral(PI/6 PI/4) (sin(x)/(1+cos(x))^2)dx

$\displaystyle \int ^{\pi/6}_{\pi/4} \left[ \frac{\sin (x)}{(1+\cos (x))^2}\right]$

Let $\displaystyle u = 1+\cos(x)$
$\displaystyle du = -\sin(x) dx$

$\displaystyle dx = -\frac{du}{\sin(x)}$

Change the limits using our substitution:

$\displaystyle u_2 = 1+\cos \left(\frac{\pi}{6}\right) = \frac{2+\sqrt3}{2}$

$\displaystyle u_1 = 1+ \cos \left(\frac{\pi}{4}\right) = \frac{2+\sqrt2}{2}$

$\displaystyle -\int ^{\frac{2+\sqrt3}{2}}_{\frac{2+\sqrt2}{2}} \left(\frac{du}{u^2}\right)$

Which should be easy enough to solve