what is the antiderivative of (5) 3^x + x^2?

Question

I tried to find the antiderivative, but i think I am making a mistake. I found (5x)(ln(3)e^ln(3)x + x^3/3

Answer 1

I assume you mean 5*3^x + x^2.

Then, the antiderivative is 5*(3^x) / (ln 3) + x^3/3 + C.

This comes from the fact that 3^x = e^(x*ln(3)), and the antiderivative of e^(ax) is e^(ax)/a + C for any nonzero constant a.

Answer 2

S [ 5(3^x) + x^2] dx = 5(3^x)/(ln 3) + (x^3)/3 + c
where S stands for the integral sign, and c is an arbitrary constant of integration.

I assume you have a problem with integrating the general power function(which appears here as 3^x)
say y = a^x
it can be shown easily by changing the base here to e that,
dy/dx = (a^x) ln a
the result for the antiderivative follows as ln a is a constant.

Answer 3

The first term should be 5*(3^x)/ln(3).