### Home > APCALC > 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 $u = 2 - e^x$.

Do not 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.

$2x f(x^2)$

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

Let $u = \sec(x)$.

$\arctan(u) + C =$
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. $\int _ { 0 } ^ { \pi / 2 } \operatorname { tan } ( \frac { x } { 2 } ) d x$

$\int_{0}^{\pi/2}\frac{\sin{(x/2)}}{\cos{(x/2)}}dx=$

Use $u$-substitution.
And do not forget about the bounds!