Question

Find the antiderivative of f (x) = 10x4 + 12.5.

Answer 1

The anti-derivative of f(x) will be:

`\int \:10x^4+12.5dx=2x^5+12.5x+C`

Explanation:

Given the function

`\:f\left(x\right)=10x^4\:+\:12.5`

Taking the anti-derivative of f(x)

`\int \left(10x^4\:+\:12.5\right)\:dx\:`

`\mathrm{Apply\:the\:Sum\:Rule}:\quad \int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx`

`\int \left(10x^4\:+\:12.5\right)\:dx\:=\int \:10x^4dx+\int \:12.5dx`

Solving

`\int 10x^4dx`

`\mathrm{Take\:the\:constant\:out}:\quad \int a\cdot f\left(x\right)dx=a\cdot \int f\left(x\right)dx`

`=10\cdot \int \:x^4dx`

`\mathrm{Apply\:the\:Power\:Rule}:\quad \int x^adx=(x^(a+1))/(a+1),\:\quad \:a\\e -1`

`=2x^5`

similarly,

`\int 12.5dx`

`\mathrm{Integral\:of\:a\:constant}:\quad \int adx=ax`

`=12.5x`

so substituting these values

`\int \left(10x^4\:+\:12.5\right)\:dx\:=\int \:10x^4dx+\int \:12.5dx`

`=2x^5+12.5x`

`=2x^5+12.5x+C` ∵ Add constant to the solution

Therefore, the anti-derivative of f(x) will be:

`\int \:10x^4+12.5dx=2x^5+12.5x+C`