Exercises - Antiderivatives

1. Find the following:

1. $\displaystyle{\int x^3 \ dx}$

2. $\displaystyle{\int x^{-4} \ dx}$

3. $\displaystyle{\int \sqrt{x} \ dx}$

4. $\displaystyle{\int \frac{1}{x} \ dx}$

5. $\displaystyle{\int \ dx}$

6. $\displaystyle{\int (x^2 + 4x) \ dx}$

7. $\displaystyle{\int (x-3)^2 \ dx}$

8. $\displaystyle{\int \frac{x-4}{\sqrt{x}} \ dx}$

9. $\displaystyle{\int \sec^2 x \ dx}$

2. If   $f\,'(x) = 3x^2-2x+5$ and the point $(2,0)$ is on the graph of $y=f\,(x)$, find $f\,(x)$.

3. Find $f\,(x)$ in each case, using the information given

1. $f\,''(x) = -\sin x, \quad f\,'(0) = -1, \quad f\,(0) = 2$

2. $f\,''(x) = -\sin x, \quad f\,'(0) = 2, \quad f\,(0) = -1$

3. $f\,''(x) = -\sin x + \cos x, \quad f\,'(0) = 2, \quad f\,(0) = 1$

4. $f\,''(x) = 6x+6, \quad f\,'(1) = 8, \quad f\,(0) = 4$

4. Evaluate the following:

1. $\displaystyle{\int \frac{4x+2\sqrt{x}}{3x^2} \ dx}$

2. $\displaystyle{\int \frac{3}{2\sin^2 t} \ dt}$

3. $\displaystyle{\int \frac{2}{e^{-x}} \ dx}$

4. $\displaystyle{\int \frac{\sin x}{\cos^2 x} \ dx}$

5. $\displaystyle{\int \frac{1+\tan^2 \theta}{5} \ d\theta}$

6. $\displaystyle{\int \frac{(1-2\sqrt{u})^2}{(2u)^3} \ du}$

7. $\displaystyle{\int \frac{4}{\pi \sec t} \ dt}$

8. $\displaystyle{\int e^{-x} (e^x \sqrt{x} + 3e^{2x}) \ dx}$

9. $\displaystyle{\int \frac{3\cos^2 3y + 3 \sin^2 3y}{2 \sqrt{y}} \ dy}$

10. $\displaystyle{\int \frac{3x}{\pi a^2} \ dt}$