Examine the following integrals. Consider the multiple tools available for evaluating integrals and use the best strategy for each part. Evaluate each integral and briefly describe your method.

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

   Hint (a):

   Before integrating, rewrite the integrand with a fractional exponent.

   Partial Answer (a):

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

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

   Hint (b):

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

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

   Hint (c):

   Undo the Power Rule.

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

   Hint (d):

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

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

   Hint (e):

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

   Answer (e):

   $2\sin(θ−1)+C$

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

   Hint (f):

   Before integrating, rewrite the integrand with a negative exponent.