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Difficult Problems

1

Solved example of Factorization

$\int\left(9+x^2\right)^{\left(2\right)\left(-1\right)}dx$
2

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

$\int\frac{1}{\left(9+x^2\right)^{2}}dx$
3

Solve the integral $\int\frac{1}{\left(9+x^2\right)^{2}}dx$ by trigonometric substitution using the substitution

$\begin{matrix}x=3\tan\left(\theta\right) \\ dx=3\sec\left(\theta\right)^2d\theta\end{matrix}$
4

Substituting in the original integral, we get

$\int\frac{3\sec\left(\theta\right)^2}{\left(9+9\tan\left(\theta\right)^2\right)^{2}}d\theta$
5

Factor by the greatest common divisor $9$

$\int\frac{3\sec\left(\theta\right)^2}{\left(9\left(1+\tan\left(\theta\right)^2\right)\right)^{2}}d\theta$
6

The power of a product is equal to the product of it's factors raised to the same power

$\int\frac{3\sec\left(\theta\right)^2}{81\left(1+\tan\left(\theta\right)^2\right)^{2}}d\theta$
7

Take $\frac{3}{81}$ out of the fraction

$\int\frac{\frac{1}{27}\sec\left(\theta\right)^2}{\left(1+\tan\left(\theta\right)^2\right)^{2}}d\theta$
8

Applying the trigonometric identity: $\tan(x)^2+1=\sec(x)^2$

$\int\frac{\frac{1}{27}\sec\left(\theta\right)^2}{\sec\left(\theta\right)^{4}}d\theta$
9

Applying the trigonometric identity: $\displaystyle\frac{1}{\sec^{n}(\theta)}=\cos^{n}(\theta)$

$\int\frac{1}{27}\cos\left(\theta\right)^{4}\sec\left(\theta\right)^2d\theta$
10

Apply the formula: $\cos\left(x\right)^n\sec\left(x\right)^m$$=\cos\left(x\right)^{\left(n-m\right)}, where x=\theta, m=2 and n=4 \int\frac{1}{27}\cos\left(\theta\right)^{2}d\theta 11 The integral of a constant by a function is equal to the constant multiplied by the integral of the function \frac{1}{27}\int\cos\left(\theta\right)^{2}d\theta 12 Apply the formula: \int\cos\left(x\right)^2dx$$=\frac{1}{2}x+\frac{1}{4}\sin\left(2x\right)$, where $x=\theta$

$\frac{1}{27}\left(\frac{1}{2}\theta+\frac{1}{4}\sin\left(2\theta\right)\right)$
13

Expressing the result of the integral in terms of the original variable

$\frac{1}{27}\left(\frac{1}{2}arctan\left(\frac{x}{3}\right)+\frac{1}{4}\sin\left(2\theta\right)\right)$
14

Using the sine double-angle identity

$\frac{1}{27}\left(\frac{1}{2}arctan\left(\frac{x}{3}\right)+\frac{1}{2}\sin\left(\theta\right)\cos\left(\theta\right)\right)$
15

Expressing the result of the integral in terms of the original variable

$\frac{1}{27}\left(\frac{1}{2}arctan\left(\frac{x}{3}\right)+\frac{\frac{3}{2}x}{9+x^2}\right)$
16

As the integral that we are solving is an indefinite integral, when we finish we must add the constant of integration

$\frac{1}{27}\left(\frac{1}{2}arctan\left(\frac{x}{3}\right)+\frac{\frac{3}{2}x}{9+x^2}\right)+C_0$

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