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### Difficult Problems

1

Solved example of integration by trigonometric substitution

$\int\sqrt{x^2+4}dx$
2

Solve the integral $\int\sqrt{x^2+4}dx$ by trigonometric substitution using the substitution

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

Substituting in the original integral, we get

$\int2\sqrt{4\tan\left(\theta\right)^2+4}\sec\left(\theta\right)^2d\theta$
4

The integral of a constant by a function is equal to the constant multiplied by the integral of the function

$2\int\sqrt{4\tan\left(\theta\right)^2+4}\sec\left(\theta\right)^2d\theta$
5

Factor by the greatest common divisor $4$

$2\int\sqrt{4\left(\tan\left(\theta\right)^2+1\right)}\sec\left(\theta\right)^2d\theta$
6

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

$2\int2\sqrt{\tan\left(\theta\right)^2+1}\sec\left(\theta\right)^2d\theta$
7

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

$2\int2\sec\left(\theta\right)^2\sec\left(\theta\right)d\theta$
8

The integral of a constant by a function is equal to the constant multiplied by the integral of the function

$4\int\sec\left(\theta\right)^2\sec\left(\theta\right)d\theta$
9

When multiplying exponents with same base you can add the exponents

$4\int\sec\left(\theta\right)^{3}d\theta$
10

Simplify the integral of secant applying the reduction formula, $\displaystyle\int\sec(x)^{n}dx=\frac{\sin(x)\sec(x)^{n-1}}{n-1}+\frac{n-2}{n-1}\int\sec(x)^{n-2}dx$

$4\left(\frac{\sec\left(\theta\right)^{2}\sin\left(\theta\right)}{2}+\frac{1}{2}\int\sec\left(\theta\right)d\theta\right)$
11

Solve the product $4\left(\frac{\sec\left(\theta\right)^{2}\sin\left(\theta\right)}{2}+\frac{1}{2}\int\sec\left(\theta\right)d\theta\right)$

$4\left(\frac{\sec\left(\theta\right)^{2}\sin\left(\theta\right)}{2}\right)+2\int\sec\left(\theta\right)d\theta$
12

Simplify the fraction

$2\sec\left(\theta\right)^{2}\sin\left(\theta\right)+2\int\sec\left(\theta\right)d\theta$
13

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

$\frac{\frac{1}{2}x\left(x^2+4\right)}{\sqrt{x^2+4}}+2\int\sec\left(\theta\right)d\theta$
14

Simplifying the fraction by $x^2+4$

$\frac{1}{2}x\sqrt{x^2+4}+2\int\sec\left(\theta\right)d\theta$
15

The integral of the secant function is given by the following formula, $\displaystyle\int\sec(x)dx=\ln\left|\sec(x)+\tan(x)\right|$

$\frac{1}{2}x\sqrt{x^2+4}+2\ln\left|\sec\left(\theta\right)+\tan\left(\theta\right)\right|$
16

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

$\frac{1}{2}x\sqrt{x^2+4}+2\ln\left|\sec\left(\theta\right)+\tan\left(\theta\right)\right|+C_0$
17

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

$\frac{1}{2}x\sqrt{x^2+4}+2\ln\left|\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right|+C_0$
18

Add fraction's numerators with common denominators: $\frac{\sqrt{x^2+4}}{2}$ and $\frac{x}{2}$

$\frac{1}{2}x\sqrt{x^2+4}+2\ln\left|\frac{\sqrt{x^2+4}+x}{2}\right|+C_0$

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