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Example Evaluate the following integral．

$\displaystyle \int (3x^{-3} + 4x^{5})dx$

Answer To find the integral, it is important to know the followings:．

1. The integral of a sum is the sum of the integrals． $\int (f(x) \pm g(x))dx = \int f(x)dx \pm \int g(x)dx$
2. The integral of a constant multiple is the constant times the integral． $\int (\alpha f(x))dx = \alpha \int f(x)$
3. Integration By Substitution． $\int f(x)dx = \int f(\Phi(t))(\Phi(t))'dt $
4. Integtation By Parts. $\int f(x)g'(x)dx = f(x)g(x) - \int f'(x)g(x)dx$
5. Basic formula for integration． $\int e^{x}dx = e^{x} + c, \int \sin{x} = -\cos{x}+c, \int \cos{x} = \sin{x} + c$

Apply these rules to the above problem, we have

$\displaystyle \int (3x^{-3} + 4x^{5})dx$ $\displaystyle =$ $\displaystyle \int 3x^3 dx + \int 4x^5dx$  
  $\displaystyle =$ $\displaystyle 3\int x^3 dx + 4\int x^5 dx$  
  $\displaystyle =$ $\displaystyle \frac{3}{4}x^4 + \frac{2}{3}x^6 + c$