### Home > CALC > Chapter 8 > Lesson 8.3.2 > Problem8-114

8-114.

Integrate the following problems without using your calculator. Be sure to change your bounds if you use $u$-substitution.

1. $\int _ { 1 } ^ { 5 } \frac { x } { \sqrt { 2 x - 1 } } d x$

Then $x=\frac{1}{2}(U+1)$ .
Let $U = 2x − 1$.
$\frac{dU}{dx}=2$
$\frac{1}{2}dU=dx$
Rewrite the integrand and don't forget to write the bounds in terms of $U$.

$=\int_1^9\frac{U+1}{4\sqrt{U}}dU=\frac{1}{4}\int_1^9(U^{1/2}+U^{-1/2})=\frac{16}{3}$

1. $\int _ { 0 } ^ { 1 } x \sqrt { 1 - x ^ { 2 } } d x$

Let $U = 1 − x^2$

1. $\int _ { 0 } ^ { 2 } \frac { x } { x ^ { 2 } + 1 } d x$

$=\int_1^5\frac{1}{2u}du=\frac{1}{2}\ln(u)\Big|_1^5=\frac{1}{2}\ln(5)$

1. $\int _ { 0 } ^ { 2 } 3 x e ^ { - x ^ { 2 } } d x$

$=\int_0^{-4}-\frac{3}{2}e^udu=\frac{3}{2}-\frac{3}{2}e^{-4}$