Calculus/Integration/Exercises

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Integration/Exercises

Integration of PolynomialsEdit

Evaluate the following:

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

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

2. \int 8x^3\, dx

2x^{4}+C

3. \int (4x^2+11x^3)\, dx

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

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

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

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

-\frac{5}{x}+C

Solutions

Indefinite IntegrationEdit

Find the general antiderivative of the following:

6. \int (\cos x+\sin x)\, dx

\sin x-\cos x+C

7. \int 3\sin x\, dx

-3\cos(x)+C

8. \int (1+\tan^2 x)\, dx

\tan x+C

9. \int (3x-\sec^2 x)\, dx

\frac{3x^{2}}{2}-\tan x+C

10. \int -e^x\, dx

-e^{x}+C

11. \int 8e^x\, dx

8e^{x}+C

12. \int \frac1{7x}\, dx

\frac{1}{7}\ln|x|+C

13. \int \frac1{x^2+a^2}\, dx

\frac{1}{a}\arctan\frac{x}{a}+C

Solutions

Integration by partsEdit

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

a. \frac{\sin^{2}x}{2}
b. -\frac{\cos^{2}x}{2}

Solutions

← Integration techniques/Numerical Approximations Calculus Area →
Integration/Exercises
Last modified on 3 July 2011, at 10:55