### Home > CALC > Chapter 11 > Lesson 11.1.3 > Problem11-33

11-33.

Examine the integrals below. Consider the multiple tools available for integrating and use the best strategy. After evaluating each integral, write a short description of your method.

1. $\int _ { 2 } ^ { 6 } \frac { 1 } { \sqrt { x - 2 } } d x$

$=\lim\limits_{a\to 2}\int_a^6(x-2)^{-1/2}dx$

1. $\int \operatorname { ln } x d x$

Use integration by parts. Let $f =\operatorname{ln}(x)$ and $dg = dx$.

1. $\int _ { 3 } ^ { t ^ { 2 } } \frac { d } { d x } ( \frac { x } { x - 2 } ) d x$

This is the integral of a derivative, so use the Fundamental Theorem of Calculus.

1. $\int _ { 1 } ^ { 2 } \frac { 1 } { 2 x ( x - 3 ) } d x$

Use partial fraction decomposition.

$\frac{1}{2x(x-3)}=\frac{a}{2x}+\frac{b}{x-3}$

$a(x-3)+b(2x)=1$

$-3a=1\text{ and }ax+2bx=0x$

$a=-\frac{1}{3}\text{ }b=\frac{1}{6}$

$\int_1^2\Big(-\frac{1}{6x}+\frac{1}{6(x-3)}\Big)dx$