Answer:
The anti-derivative of f(x) will be:
[tex]\int \:10x^4+12.5dx=2x^5+12.5x+C[/tex]
Step-by-step explanation:
Given the function
[tex]\:f\left(x\right)=10x^4\:+\:12.5[/tex]
Taking the anti-derivative of f(x)
[tex]\int \left(10x^4\:+\:12.5\right)\:dx\:[/tex]
[tex]\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[/tex]
[tex]\int \left(10x^4\:+\:12.5\right)\:dx\:=\int \:10x^4dx+\int \:12.5dx[/tex]
Solving
[tex]\int 10x^4dx[/tex]
[tex]\mathrm{Take\:the\:constant\:out}:\quad \int a\cdot f\left(x\right)dx=a\cdot \int f\left(x\right)dx[/tex]
[tex]=10\cdot \int \:x^4dx[/tex]
[tex]\mathrm{Apply\:the\:Power\:Rule}:\quad \int x^adx=\frac{x^{a+1}}{a+1},\:\quad \:a\ne -1[/tex]
[tex]=2x^5[/tex]
similarly,
[tex]\int 12.5dx[/tex]
[tex]\mathrm{Integral\:of\:a\:constant}:\quad \int adx=ax[/tex]
[tex]=12.5x[/tex]
so substituting these values
[tex]\int \left(10x^4\:+\:12.5\right)\:dx\:=\int \:10x^4dx+\int \:12.5dx[/tex]
[tex]=2x^5+12.5x[/tex]
[tex]=2x^5+12.5x+C[/tex] ∵ Add constant to the solution
Therefore, the anti-derivative of f(x) will be:
[tex]\int \:10x^4+12.5dx=2x^5+12.5x+C[/tex]
