Calculus
Integration Techniques
Using the substitution \(\ln x=t,\) which of the following is equal to \[\int_{e}^{e^{6}} \frac{3(\ln x)^2}{x} dx ?\]
Evaluate \(\displaystyle{\int_0^{3}\frac{12x+6}{x^2+x+1}dx.}\)
Using the substitution \(u=\sin x,\) evaluate \(\displaystyle{\int_0^\frac{\pi}{2}\sin^{10} x\cos xdx.}\)
Evaluate \(\displaystyle{\int_0^\frac{\pi}{32}3\tan8xdx.}\)
Using the substitution \(u=e^{x^2},\) evaluate \(\displaystyle{\int_0^{3}12x e^{x^2}dx.}\)