Integrate the following $\int \sqrt{x^{2} + 81} d x$ $int sqrt(x^2+81) dx$ ?

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Integrate the following $\int \sqrt{x^{2} + 81} d x$ $int sqrt(x^2+81) dx$ ?

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Answer 1 · 2017-02-07T14:39:45

$\frac{x \sqrt{x^{2} + 81}}{2} + \frac{81}{2} ln ∣ ∣ ∣ \frac{x + \sqrt{x^{2} + 81}}{9} ∣ ∣ ∣ + C$ $(xsqrt(x^2 + 81))/2 + 81/2ln|(x + sqrt(x^2 + 81))/9| + C$

Explanation:

Use trig substitution. Let $x = 9 tan θ$ $x= 9tantheta$ . Then $d x = 9 {sec}^{2} θ d θ$ $dx = 9sec^2theta d theta$ .

$\Rightarrow \int \sqrt{{(9 tan θ)}^{2} + 81} \cdot 9 {sec}^{2} θ d θ$ $=>intsqrt((9tantheta)^2 + 81) * 9sec^2theta d theta$

$\Rightarrow \int \sqrt{81 {tan}^{2} θ + 81} \cdot 9 {sec}^{2} θ d θ$ $=>intsqrt(81tan^2theta + 81) * 9sec^2theta d theta$

$\Rightarrow \int \sqrt{81 ({tan}^{2} θ + 1)} \cdot 9 {sec}^{2} θ d θ$ $=>intsqrt(81(tan^2theta +1)) * 9sec^2theta d theta$

$\Rightarrow \int \sqrt{81 {sec}^{2} θ} \cdot 9 {sec}^{2} θ d θ$ $=>intsqrt(81sec^2theta) * 9sec^2theta d theta$

$\Rightarrow \int 9 sec θ \cdot 9 {sec}^{2} θ d θ$ $=>int 9sectheta * 9sec^2theta d theta$

$\Rightarrow \int 81 {sec}^{3} θ d θ$ $=>int 81sec^3theta d theta$

$\Rightarrow 81 \int {sec}^{3} θ d θ$ $=>81int sec^3theta d theta$

This is a relatively known integral. The proof can be found here.

$\Rightarrow \frac{81}{2} sec θ tan θ + \frac{81}{2} ln | sec θ + tan θ | + C$ $=>81/2secthetatantheta+ 81/2ln|sectheta+ tantheta| + C$

We know from our initial substitution that $tan θ = \frac{x}{9}$ $tantheta = x/9$ . This means the hypotenuse of the triangle would have a hypotenuse of $\sqrt{x^{2} + 81}$ $sqrt(x^2 + 81)$ .

$\Rightarrow \frac{81}{2} (\frac{\sqrt{x^{2} + 81}}{9}) (\frac{x}{9}) + \frac{81}{2} ln ∣ ∣ ∣ \frac{\sqrt{x^{2} + 81}}{9} + \frac{x}{9} ∣ ∣ ∣ + C$ $=>81/2(sqrt(x^2 + 81)/9)(x/9) + 81/2ln|sqrt(x^2 + 81)/9 + x/9| + C$

$\Rightarrow \frac{x \sqrt{x^{2} + 81}}{2} + \frac{81}{2} ln ∣ ∣ ∣ \frac{x + \sqrt{x^{2} + 81}}{9} ∣ ∣ ∣ + C$ $=>(xsqrt(x^2 + 81))/2 + 81/2ln|(x + sqrt(x^2 + 81))/9| + C$

Hopefully this helps!

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Integrate the following $\int \sqrt{x^{2} + 81} d x$ $int sqrt(x^2+81) dx$ ?

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