CPM Homework Help : CALC Problem 5-10

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5-10.

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

$\int _ { 1 } ^ { 2 } ( 15 x ^ { 4 } + \sqrt [ 3 ] { x } ) d x$
Hint (a):
Before integrating, rewrite the integrand with a fractional exponent.
Partial Answer (a):
$\int_1^2(15x^4+x^{1/3})dx=3x^5+\frac{3}{4}x^{4/3}\Big|_1^2=?$

$\int _ { 0 } ^ { \pi / 3 } 4 \operatorname { sin } ( y ) d y$
Hint (b):
$\int\sin(x)dx=-\cos(x)+C$

$\int _ { - 2 } ^ { 3 } ( 3 x ^ { 2 } + 1 ) d x$
Hint (c):
Undo the Power Rule.

$\int _ { - 2 } ^ { 2 } ( \sqrt { 4 - x ^ { 2 } } ) d x$
Hint (d):
Do NOT use a calculator!
This function is a graph of a semicircle.
Calculate the area using geometry.

$\int 2 \operatorname { cos } ( \theta - 1 ) d \theta$
Hint (e):
Determine the antiderivative by shifting and stretching the antiderivative of $y =\operatorname{cos}(x)$ .
Answer (e):
$2\operatorname{sin}(θ-1) + C$

$\int \frac { 2 } { x ^ { 2 } } d x$
Hint (f):
Before integrating, rewrite the integrand with a negative exponent.

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