  ### Home > CALC > Chapter 8 > Lesson 8.1.3 > Problem8-32

8-32.

No calculator! Integrate.

1. $\int _ { 0 } ^ { 1 } \frac { e ^ { x } } { ( 2 - e ^ { x } ) ^ { 2 } } d x$

Let $U = 2 − ex$.

Don't forget to rewrite the bounds in terms of $u$.

1. $\frac { d } { d x } \int _ { 4 } ^ { x ^ { 2 } } f ( x ) d x$

The derivative of an INDEFINITE integral, is the original function (FTC, pt.1) But the derivative of a DEFINITE integral is the original function times the derivative of the bounds.

$2xf(x^2)$

1. $\int \frac { \operatorname { sec } x \operatorname { tan } x } { 1 + \operatorname { sec } ^ { 2 } x } d x$

Let $u =\operatorname{sec}(x)$.

$\operatorname{arctan}(u) + C =$
Now rewrite the answer in terms of $x$.

1. $\int \frac { x } { \sqrt { 1 - x ^ { 4 } } } d x$

Recall that $x^4 = (x^2)^2$.

1. $\int _ { 0 } ^ { \pi / 2 } \operatorname { tan } ( \frac { x } { 2 } ) d x$

$\int_{0}^{\frac{\pi }{2}}\frac{\text{sin}\frac{x}{2}}{\text{cos}\frac{x}{2}}dx=$

Use $u$-substitution.
And don't forget about the bounds! 