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:

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}

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

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{eq}\displaystyle \eqalign{ & 1)\,\,\int {x\sin {x^2}dx}. \cr & {\text{Let,}}\,{\text{ }}u = {x^2}\, \Rightarrow \,\,du = 2xdx \cr & \int...

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Using Integration By Parts

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Chapter 13 / Lesson 7
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Learn how to use and define integration by parts. Discover the integration by parts rule and formula. Learn when and how to use integration by parts with examples.


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