Home > CALC > Chapter 5 > Lesson 5.1.1 > Problem5-10

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.

1. $\int _ { 1 } ^ { 2 } ( 15 x ^ { 4 } + \sqrt [ 3 ] { x } ) d x$

Before integrating, rewrite the integrand with a fractional exponent.

$\int_1^2(15x^4+x^{1/3})dx=3x^5+\frac{3}{4}x^{4/3}\Big|_1^2=?$

1. $\int _ { 0 } ^ { \pi / 3 } 4 \operatorname { sin } ( y ) d y$

$\int\sin(x)dx=-\cos(x)+C$

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

Undo the Power Rule.

1. $\int _ { - 2 } ^ { 2 } ( \sqrt { 4 - x ^ { 2 } } ) d x$

Do NOT use a calculator!
This function is a graph of a semicircle.
Calculate the area using geometry.

1. $\int 2 \operatorname { cos } ( \theta - 1 ) d \theta$

Determine the antiderivative by shifting and stretching the antiderivative of $y =\operatorname{cos}(x)$.

$2\operatorname{sin}(θ-1) + C$

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

Before integrating, rewrite the integrand with a negative exponent.