Exam 4

1. Find the following indefinite integrals. (8% each)

   1. $$\int (5x + 7) \, dx$$

   2. $$\int (3 e^{x} + 2 \sin x) \, dx$$

   3. $$\int \frac{1}{\cos^{2} x} \, dx$$

   4. $$\int \left( \frac{x+1}{x} \right) \, dx$$

   5. $$\int \left( 3 \cos x + 3 \sqrt{x} \right) \, dx$$

   6. $$\int x \cos (x^{2}) \, dx$$

   7. $$\int \frac{1}{x+5} \, dx$$

2. Find the following derivatives. (8% each)

   1. $$\frac{d}{dx} \int_{0}^{x} \cos (t^{2}) \, dt$$

   2. $$\frac{d}{dx} \int_{x}^{1} \ln t \, dt$$

   3. $$\frac{d}{dx} \int_{0}^{x^{2}} \frac{\sin t}{t} \, dt$$

3. Find the area enclosed by $y=x(x-\pi)$ and $y= \sin x$ . (10%)
4. If $f$ is continuous and $\int_{0}^{4} f(x) \, d x = 10$ , find $\int_{0}^{2} f(2 x) \, d x$ . (10%)