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8-114.

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

$\int _ { 1 } ^ { 5 } \frac { x } { \sqrt { 2 x - 1 } } d x$
Hint (a):
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 $U$ .
Answer (a):
$=\int_1^9\frac{U+1}{4\sqrt{U}}dU=\frac{1}{4}\int_1^9(U^{1/2}+U^{-1/2})=\frac{16}{3}$

$\int _ { 0 } ^ { 1 } x \sqrt { 1 - x ^ { 2 } } d x$
Hint (b):
Let $U = 1 − x^2$

$\int _ { 0 } ^ { 2 } \frac { x } { x ^ { 2 } + 1 } d x$
Answer (c):
$=\int_1^5\frac{1}{2u}du=\frac{1}{2}\ln(u)\Big|_1^5=\frac{1}{2}\ln(5)$

$\int _ { 0 } ^ { 2 } 3 x e ^ { - x ^ { 2 } } d x$
Answer (d):
$=\int_0^{-4}-\frac{3}{2}e^udu=\frac{3}{2}-\frac{3}{2}e^{-4}$

Compute without a calculator

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