Calculus/Integration/Solutions

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[edit] Solutions to Set One

[edit] Solutions to Set Two

  1. \frac{1}{13} x^{13}+C \,
  2. 2x^4+C \,
  3. \frac{4}{3} x^3+\frac{11}{4} x^4 +C \,
  4. \frac{31}{33}x^{33}+x^4 - \frac{9}{5}x^5+ C \,
  5. -5 x^{-1}+C \,

[edit] Solutions to Set Three

1. \int (\cos{x}+\sin{x})\, dx

=\int \cos{x}\, dx + \int \sin {x}\, dx
=\sin{x} - \cos{x}\ + C

2. \int 3\sin{x}\, dx

=3 \times\int \sin{x}\, dx
=-3 \cos{x} + C\

3. \int 1+\tan^2{x}\, dx

=\int \sec^2{x}\, dx
= tanx + C

4. {3x^2 \over 2} - \tan{x}+ C

5. -e^x +\ C

6. 8e^x +\ C

7. {1 \over 7} \ln \left| x \right| +\ C

8. with the substitution x = atanθ, we have dx = a \sec^2\theta\, d\theta, and x2 + a2 = a2sec2θ, so that

\int \frac1{x^2+a^2}\, dx
=\int \frac{a \sec^2\theta\, d\theta}{a^2 \sec^2\theta}
=\int\frac1{a}\, d\theta
=\frac1{a}\theta + C
=\frac1{a}\arctan \frac{x}{a} + C
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