Calculus/Integration/Exercises

 ← Integration techniques/Numerical Approximations Calculus Area → Integration/Exercises

Integration of Polynomials

Evaluate the following:

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

${\displaystyle {\frac {x^{5}}{5}}-{\frac {4x^{3}}{3}}+4x+C}$

2. ${\displaystyle \int 8x^{3}dx}$

${\displaystyle 2x^{4}+C}$

3. ${\displaystyle \int (4x^{2}+11x^{3})dx}$

${\displaystyle {\frac {4x^{3}}{3}}+{\frac {11x^{4}}{4}}+C}$

4. ${\displaystyle \int (31x^{32}+4x^{3}-9x^{4})dx}$

${\displaystyle {\frac {31x^{33}}{33}}+x^{4}-{\frac {9x^{5}}{5}}+C}$

5. ${\displaystyle \int 5x^{-2}\,dx}$

${\displaystyle -{\frac {5}{x}}+C}$

Solutions

Indefinite Integration

Find the general antiderivative of the following:

6. ${\displaystyle \int {\bigl (}\cos(x)+\sin(x){\bigr )}dx}$

${\displaystyle \sin(x)-\cos(x)+C}$

7. ${\displaystyle \int 3\sin(x)dx}$

${\displaystyle -3\cos(x)+C}$

8. ${\displaystyle \int {\bigl (}1+\tan ^{2}(x){\bigr )}dx}$

${\displaystyle \tan(x)+C}$

9. ${\displaystyle \int {\bigl (}3x-\sec ^{2}(x){\bigr )}dx}$

${\displaystyle {\frac {3x^{2}}{2}}-\tan(x)+C}$

10. ${\displaystyle \int -e^{x}\,dx}$

${\displaystyle -e^{x}+C}$

11. ${\displaystyle \int 8e^{x}\,dx}$

${\displaystyle 8e^{x}+C}$

12. ${\displaystyle \int {\frac {dx}{7x}}}$

${\displaystyle {\frac {\ln |x|}{7}}+C}$

13. ${\displaystyle \int {\frac {dx}{x^{2}+a^{2}}}}$

${\displaystyle {\frac {arctan{\bigl (}{\tfrac {x}{a}}{\bigr )}}{a}}+C}$

Solutions

Integration by parts

14. Consider the integral ${\displaystyle \int \sin(x)\cos(x)dx}$ . Find the integral in two different ways. (a) Integrate by parts with ${\displaystyle u=\sin(x)}$ and ${\displaystyle v'=\cos(x)}$ . (b) Integrate by parts with ${\displaystyle u=\cos(x)}$ and ${\displaystyle v'=\sin(x)}$ . Compare your answers. Are they the same?

a. ${\displaystyle {\frac {\sin ^{2}(x)}{2}}}$
b. ${\displaystyle -{\frac {\cos ^{2}(x)}{2}}}$

Solutions

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