1) Find {eq}\displaystyle \int x \sin(x^2) \ dx. {/eq}

2) Find {eq}\displaystyle \int e^x \cot(e^x) \ dx. {/eq}

3) Find {eq}\displaystyle \int \sin^{-1} x \ dx. {/eq}

Question:

Concept-related to Integration substitution

Integration by Substitution is nothing but the reverse chain rule and it is a method to find an integral, but only when it can be set up in a special way.

In Integration substitution a given integral {eq}\int {f(x)dx} {/eq} can be transformed into another form by changing the independent variable x to t,

{eq}\displaystyle \int {f(x)dx} = \int {f\left( {g(t)} \right)} \frac{{dg}}{{dt}}dt. {/eq}

For, {eq}\displaystyle x = g(t)\,\, \Rightarrow \,dx = \frac{{dg}}{{dt}}dt. {/eq}

Let {eq}u{/eq} and {eq}v{/eq} be functions of {eq}x{/eq}, then integration by parts states;

{eq}\displaystyle \int {uvdx = u\int {vdx - \int {u'} \left( {\int {vdx} } \right)dx} }. {/eq}

Answer and Explanation: 1