Question

How to solve this indefinite integral? ​

Answer 1

`\displaystyle\int(3x)/(\cos^2(2x^2))\,\mathrm dx`

Substitute
`y=2x^2`, so that
`\mathrm dy=4x\,\mathrm dx`:

`\displaystyle\int(3x)/(\cos^2(2x^2))\,\mathrm dx=\frac34\int(4x)/(\cos^2(2x^2))\,\mathrm dx=\frac34\int(\mathrm dy)/(\cos^2y)`

Then

`\frac1{\cos^2y}=\sec^2y=(\mathrm d)/(\mathrm dy)[\tan y]`

so that the integral wrt
`y` comes out to be

`\displaystyle\frac34\int\sec^2y\,\mathrm dy=\frac34\tan y+C`

Replace
`y` to solve for the integral wrt
`x`:

`\displaystyle\int(3x)/(\cos^2(2x^2))\,\mathrm dx=\boxed{\frac34\tan(2x^2)+C}`