Question

What is `int x^4 + 9 x^3 -8 x^2 -3 x + 5 dx`?

Answer 1

`1/5x^5+9/4x^4-8/3x^3-3/2x^2+5x+C`

Explanation:

Use the rule:

`intx^ndx=(x^(n+1))/(n+1)+C`

Also, recall that multiplicative constants (like the `9` in `9x^3`) can be brought outside of the integral, but just stay and are multiplied in, and that various antiderivatives can be added.

Also, the antiderivative of a constant, like `5`, is simply `5x`. This is true since `5=5x^0`, to which the original rule can be applied.

The antiderivative of the given function is:

`x^(4+1)/(4+1)+(9x^(3+1))/(3+1)-(8x^(2+1))/(2+1)-(3x^(1+1))/(1+1)+(5x^(0+1))/(0+1)+C`

`=1/5x^5+9/4x^4-8/3x^3-3/2x^2+5x+C`