Question

Find the anti derivative of (x)^(1/2)(6 + 5x)

I'm completely lost with these questions can some one show me the step by step method?

An explanation would also be apprciated but I dont expect one

Answer 1

http://www.khanacademy.org/math/calculus/v/the-indefinite-integral-or-anti-derivative

Answer 2

Yeah that video isn't helpful I know how to take the antiderivative I just dont know how to take the anti derivative of products

Answer 3

Set
u = (x)^(1/2)
dx/2(x)^(1/2) = dt

What do I do next?

Answer 4

We have
\[\int x^{\frac{1}{2}}(6+5x)dx\]

Just mulitply \(x^{\frac{1}{2}}\) to all the terms
We get
\[\large \int (6x^{\frac{1}{2}}+5x^{\frac{3}{2}})dx\]
We know the integral of
\[\int x^n dx= \frac{x^{n+1}}{n} when\ n\ne -1\]
so here we'll use this
We get
\[\large \frac{6x^{(\frac{1}{2}+1)}}{\frac{1}{2}+1}+\frac{5x^{\frac{3}{2}+1}}{\frac{3}{2}+1}+C\]
We get finally
\[\large \int x^{\frac{1}{2}}(6+5x)dx= \frac{6x^{\frac{3}{2}}}{\frac{3}{2}}+\frac{5x^{\frac{5}{2}}}{\frac{5}{2}}+C\]