Question

How do you integrate the following using the appropriate technique?

integral of: inverse sin(x) + 3/x^(4/3) - (sec^2(x)*tan(x)) dx

integral of: sqrt(x)(x^-2 - 3x) dx

integral of: lnx(3x/x^2) dx

Answer 1

\[\large \int\limits \color{orangered}{\arcsin x} +\color{royalblue}{3x^{-4/3}}-\color{brown}{\sec^2x \tan x} \; dx\]Ok the only part of this that's going to be rather difficult is the arcsine.

Do you understand how to integrate the middle term, `the blue one`?

Answer 2

By using the Power Rule for Integration*

Answer 3

Yes. I understand the blue part.

Answer 4

For the orange part, we'll need to do integration by parts.
We're going to be sneaky and instead of taking the integral of arcsine, we'll set it as our `u` so we end up differentiating it instead.

\[\large u=\arcsin x \qquad \qquad dv=dx\]This will be our `u` and `dv`.
Now let's find our `du` and `v`.
\[\large du=\frac{1}{\sqrt{1-x^2}}dx \qquad \qquad v=x\]