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Solve the integral I = \int \frac {x}{{(1-x^2)}^{3/2}} dx

Question:

Solve the integral {eq}I = \int \frac {x}{{(1-x^2)}^{3/2}} dx {/eq}

Indefinite integral

We guess the appropriate change of variable and transform the given integral into a basic integral of type {eq}\int x^n dx,~n\neq -1 {/eq}. Have in mind that since we work with an indefinite integral, the real constant {eq}C {/eq} appears. The elementary properties of an integral also used such as homogenity.

Answer and Explanation:

Using the substitute {eq}t=1-x^2 {/eq} (which implies {eq}~dt = -2xdx {/eq} , e.g. {eq}~~xdx = -\frac{1}{2}dt {/eq}) , we find

$$\begin{align} I &= \int \frac {x}{{(1-x^2)}^{3/2}} dx\\ &= -\frac{1}{2} \int \frac {dt}{t^{3/2}} \\ &= -\frac{1}{2}\cdot (-2)t^{-\frac{1}{2}} +C \\ &= \frac{1}{\sqrt{t}} + C\\ &=\frac{1}{\sqrt{1-x^2}}+C \end{align} $$

where {eq}C {/eq} is a real constant.


Learn more about this topic:

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Indefinite Integral: Definition, Rules & Examples

from Calculus: Tutoring Solution

Chapter 7 / Lesson 14
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