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Integration Techniques

Writing an integral down is only the first step. A toolkit of techniques can help find its value, from substitutions to trigonometry to partial fractions to differentiation.

Given Substitution

         

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 \(\displaystyle{\int_0^{3}12x e^{x^2}dx.}\)

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