Integration by trigonometric substitution Calculator
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2
Solve the integral $\int\sqrt{4+x^2}dx$ 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
$\int2\sqrt{4+4\tan\left(\theta\right)^2}\sec\left(\theta\right)^2d\theta$
4
Take the constant out of the integral
$2\int\sqrt{4+4\tan\left(\theta\right)^2}\sec\left(\theta\right)^2d\theta$
5
Factor by the greatest common divisor $4$
$2\int\sqrt{4\left(1+1\tan\left(\theta\right)^2\right)}\sec\left(\theta\right)^2d\theta$
6
Any expression multiplied by $1$ is equal to itself
$2\int\sqrt{4\left(1+\tan\left(\theta\right)^2\right)}\sec\left(\theta\right)^2d\theta$
7
The power of a product is equal to the product of it's factors raised to the same power
$2\int2\sec\left(\theta\right)^2\sqrt{1+\tan\left(\theta\right)^2}d\theta$
8
Applying the trigonometric identity: $\tan(x)^2+1=\sec(x)^2$
$2\int2\sec\left(\theta\right)^2\sec\left(\theta\right)d\theta$
9
When multiplying exponents with same base you can add the exponents
$2\int2\sec\left(\theta\right)^{3}d\theta$
10
Take the constant out of the integral
$4\int\sec\left(\theta\right)^{3}d\theta$
11
Simplify the integral of secant applying the reduction formula, $\displaystyle\int\sec(x)^{n}dx=\frac{\sin(x)\sec(x)^{n-1}}{n-1}+\frac{n-2}{n-1}\int\sec(x)^{n-2}dx$
$4\left(\frac{1}{2}\int\sec\left(\theta\right)d\theta+\frac{\sec\left(\theta\right)^{2}\sin\left(\theta\right)}{2}\right)$
12
Expressing the result of the integral in terms of the original variable
$4\left(\frac{1}{2}\int\sec\left(\theta\right)d\theta+\frac{\frac{x\left(\frac{\sqrt{4+x^2}}{2}\right)^{2}}{\sqrt{4+x^2}}}{2}\right)$
13
Simplifying the fraction
$4\left(\frac{1}{2}\int\sec\left(\theta\right)d\theta+\frac{x\left(\frac{\sqrt{4+x^2}}{2}\right)^{2}}{2\sqrt{4+x^2}}\right)$
14
Taking out the constant $2$ from the fraction's denominator
$4\left(\frac{1}{2}\int\sec\left(\theta\right)d\theta+\frac{1}{2}\cdot\frac{x\left(\frac{\sqrt{4+x^2}}{2}\right)^{2}}{\sqrt{4+x^2}}\right)$
15
The integral of the secant function is given by the following formula, $\displaystyle\int\sec(x)dx=\ln\left|\sec(x)+\tan(x)\right|$
$4\left(\frac{1}{2}\ln\left(\tan\left(\theta\right)+\sec\left(\theta\right)\right)+\frac{1}{2}\cdot\frac{x\left(\frac{\sqrt{4+x^2}}{2}\right)^{2}}{\sqrt{4+x^2}}\right)$
16
Expressing the result of the integral in terms of the original variable
$4\left(\frac{1}{2}\ln\left(\frac{x}{2}+\frac{\sqrt{4+x^2}}{2}\right)+\frac{\frac{1}{2}x\left(\frac{\sqrt{4+x^2}}{2}\right)^{2}}{\sqrt{4+x^2}}\right)$
17
Add fraction's numerators with common denominators: $\frac{\sqrt{4+x^2}}{2}$ and $\frac{x}{2}$
$4\left(\frac{1}{2}\ln\left(\frac{x+\sqrt{4+x^2}}{2}\right)+\frac{\frac{1}{2}x\left(\frac{\sqrt{4+x^2}}{2}\right)^{2}}{\sqrt{4+x^2}}\right)$
18
Multiply $\left(\frac{\frac{1}{2}x\left(\frac{\sqrt{4+x^2}}{2}\right)^{2}}{\sqrt{4+x^2}}+\frac{1}{2}\ln\left(\frac{x+\sqrt{4+x^2}}{2}\right)\right)$ by $4$
$2\ln\left(\frac{x+\sqrt{4+x^2}}{2}\right)+4\frac{\frac{1}{2}x\left(\frac{\sqrt{4+x^2}}{2}\right)^{2}}{\sqrt{4+x^2}}$
19
The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$
$2\ln\left(\frac{x+\sqrt{4+x^2}}{2}\right)+4\frac{\frac{1}{2}x\left(\frac{4+x^2}{4}\right)}{\sqrt{4+x^2}}$
$2\ln\left(\frac{x+\sqrt{4+x^2}}{2}\right)+4\frac{\frac{1}{8}x\left(4+x^2\right)}{\sqrt{4+x^2}}$
21
Simplifying the fraction by $4+x^2$
$2\ln\left(\frac{x+\sqrt{4+x^2}}{2}\right)+4\frac{\frac{1}{8}x}{\frac{1}{\sqrt{4+x^2}}}$
22
Simplifying the fraction
$2\ln\left(\frac{x+\sqrt{4+x^2}}{2}\right)+4\cdot \frac{1}{8}\sqrt{4+x^2}x$
23
Multiply $\frac{1}{8}$ times $4$
$2\ln\left(\frac{x+\sqrt{4+x^2}}{2}\right)+\frac{1}{2}\sqrt{4+x^2}x$
24
The logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator
$2\left(\ln\left(2\right)\left(-1\right)+\ln\left(x+\sqrt{4+x^2}\right)\right)+\frac{1}{2}\sqrt{4+x^2}x$
25
Calculating the natural logarithm of $2$
$2\left(\frac{4}{\sqrt{3}}\left(-1\right)+\ln\left(x+\sqrt{4+x^2}\right)\right)+\frac{1}{2}\sqrt{4+x^2}x$
26
Multiply $-1$ times $\frac{4}{\sqrt{3}}$
$2\left(\ln\left(x+\sqrt{4+x^2}\right)-\frac{4}{\sqrt{3}}\right)+\frac{1}{2}\sqrt{4+x^2}x$
27
Multiply $\left(\ln\left(x+\sqrt{4+x^2}\right)+-\frac{4}{\sqrt{3}}\right)$ by $2$
$\frac{1}{2}\sqrt{4+x^2}x-\frac{17}{\sqrt{3}}+2\ln\left(x+\sqrt{4+x^2}\right)$
28
Add the constant of integration
$\frac{1}{2}\sqrt{4+x^2}x-\frac{17}{\sqrt{3}}+2\ln\left(x+\sqrt{4+x^2}\right)+C_0$