Examine the following integrals. Consider the multiple tools available for evaluating integrals and use the best strategy for each part. Evaluate the definite integrals and state the strategies that you used. For the indefinite integrals, find the antiderivative function, if you can.

1. $\int _ { - \pi } ^ { \pi } \operatorname { cos } ( x ) d x$

   Hint (a):

   To make this problem quicker to solve, remember that
   $y=\cos x$ is an even function.

   Answer (a):

   $=2\int_{0}^{\pi}\text{cos}xdx=2\left ( \text{sin}x\left|\begin{matrix} \pi \\ 0 \end{matrix}\right. \right )=2[\text{sin}\pi -\text{sin}0]=0$

2. $\int ( 5 \sqrt { y } - \operatorname { sin } ( y ) ) d y$

   Hint (b):

   Notice that this is an indefinite integral. Don't forget the $+C$.

3. $\int _ { 1 } ^ { 2.7183 } \frac { 1 } { x } d x$

   Hint (c):

   $2.7183 ≈ e$ (Euler's number) and the derivative of $\ln\left(x\right)$ is $x^{−1}$.

4. $\int ( \operatorname { sin } ^ { 2 } ( x ) + \operatorname { cos } ^ { 2 } ( x ) ) d x$

   Hint (d):

   Think! Trig identity. Simplify the integrand before you integrate.

5. $\int \frac { e ^ { x } } { x } d x$