Question

Please help!!! It would be highly appreciated!!!

Answer 1

-x⁵ +3x² +3x +C

Explanation:

You want the indefinite integral of (-5x⁴ +6x +3).

Power rule

The power rule for integration is the reverse of the power rule for derivatives.

`\begin{array}{ll}\text{Derivative:}&(d(x^n))/(dx)=n\cdot x^(n-1)\\\\\text{Integral:}&\displaystyle\int {x^n}\,dx=(x^(n+1))/(n+1)\end{array}`

Application

An indefinite integral represents a family of functions, each with the same derivative. In general, they may differ by a constant. That constant is necessary to signify the functions that will have the given expression as their derivative.

`\displaystyle\int {(-5x^4+6x+3)}\,dx=-5(x^(4+1))/(4+1)+6(x^(1+1))/(1+1)+3(x^(0+1))/(0+1)=\boxed{-x^5+3x^2+3x+C}`