Calculus/Integration/Exercises

< Calculus‎ | Integration

Integration of PolynomialsEdit

Evaluate the following:

1. \int (x^2-2)^2dx

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

2. \int 8x^3dx

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\bigl(\cos(x)+\sin(x)\bigr)dx

\sin(x)-\cos(x)+C

7. \int 3\sin(x)dx

-3\cos(x)+C

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

\tan(x)+C

9. \int\bigl(3x-\sec^2(x)\bigr)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\frac{dx}{7x}

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

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

Failed to parse (lexing error): \frac{arctan\bigl(\tfrac{x}{a}\bigr)}{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

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