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For the following integral, find an appropriate trigonometric substitution of the form x = f(t)...

Question:

For the following integral, find an appropriate trigonometric substitution of the form x = f(t) to simplify the integral. {eq}\displaystyle \int((4x^2 - 3)^{1.5}) dx {/eq}

x =

Trigonometric Substitution:

If an integral consists of the form {eq}\displaystyle \sqrt{b^2x^2-a^2 } {/eq}, then the trigonometric substitution {eq}x=\displaystyle \frac{a}{b}\sec t {/eq} is required to solve it, assuming other integral techniques are not permissible.

Other forms wherein trigonometric substitutions are applied are {eq}\displaystyle \sqrt{a^2-b^2x^2 } {/eq} and {eq}\displaystyle \sqrt{a^2+b^2x^2 } {/eq}.

Answer and Explanation:

Rewriting {eq}\displaystyle \int((4x^2 - 3)^{1.5}) \ \mathrm{d}x {/eq} to determine values of {eq}a {/eq} and {eq}b {/eq}:

{eq}\begin{align*} \displaystyle \int((4x^2 - 3)^{1.5}) \ \mathrm{d}x& = \displaystyle \int(\sqrt{4x^2 - 3})^3 \ \mathrm{d}x\\ & = \displaystyle \int\left(\sqrt{2^2x^2 - (\sqrt{3})^2}\right)^3 \ \mathrm{d}x\\ \end{align*} {/eq}

It's clear it has the form {eq}\displaystyle \sqrt{b^2x^2-a^2 } {/eq}, such that {eq}a=\sqrt{3} {/eq} and {eq}b=2 {/eq}, so we need the trigonometric substitution {eq}x=\displaystyle \frac{a}{b}\sec t {/eq}.

As {eq}a=\sqrt{3} {/eq} and {eq}b=2 {/eq}, the appropriate trigonometric substitution is {eq}x=\displaystyle \frac{\sqrt{3}}{2}\sec t {/eq}.


Learn more about this topic:

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How to Use Trigonometric Substitution to Solve Integrals

from Math 104: Calculus

Chapter 13 / Lesson 11
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