Interesting Integration

$\large \int \sqrt{x + \sqrt{x^2 + 3x}} \, dx = \, ?$

#Calculus

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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Comments

Brief solution:

Let $y = x + \sqrt{x^2 + 3x}$ , then $(y-x)^2 = x^2 + 3x \Rightarrow x = \frac{y^2}{2y+3}$ .

The integral becomes $\displaystyle \int \sqrt y \cdot \dfrac{2y^2 + 6y}{(2y+3)^2} \, dy$ . Then let $z = \sqrt y$ , the integral transforms to $\displaystyle 4 \int \dfrac{z^6 + 3z^4}{(2z^2+3)^2} \, dz$ .

By long division, this integral simplifies to $\displaystyle 4 \int \left( \dfrac14 z^2- \dfrac98 \cdot \dfrac1{2z^2+3} + \dfrac{27}8 \dfrac1{(2z^2+3)^2} \right) \, dz$ .

We just need to integral the remaining rational functions but using a trigonometric substitution, $z = \dfrac{\sqrt3}{\sqrt 2} \tan w$ .

Can you finish it from here?

Pi Han Goh - 3 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$