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Solved example of integration by trigonometric substitution
$\int\sqrt{x^2+4}dx$
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We can solve the integral $\int\sqrt{x^2+4}dx$ by applying integration method of trigonometric substitution using the substitution
$x=2\tan\left(\theta \right)$
Intermediate steps
The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function
$2\frac{d}{d\theta}\left(\tan\left(\theta \right)\right)$
The derivative of the tangent of a function is equal to secant squared of that function times the derivative of that function, in other words, if ${f(x) = tan(x)}$, then ${f'(x) = sec^2(x)\cdot D_x(x)}$
$2\sec\left(\theta \right)^2\frac{d}{d\theta}\left(\theta \right)$
The derivative of the linear function is equal to $1$
$2\sec\left(\theta \right)^2$
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Now, in order to rewrite $d\theta$ in terms of $dx$, we need to find the derivative of $x$. We need to calculate $dx$, we can do that by deriving the equation above
$dx=2\sec\left(\theta \right)^2d\theta$
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Substituting in the original integral, we get
$\int2\sqrt{4\tan\left(\theta \right)^2+4}\sec\left(\theta \right)^2d\theta$
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Factor by the greatest common divisor $4$
$\int2\sqrt{4\left(\tan\left(\theta \right)^2+1\right)}\sec\left(\theta \right)^2d\theta$
Intermediate steps
$\int2\sqrt{2^2\tan\left(\theta \right)^2+4}\sec\left(\theta \right)^2d\theta$
Calculate the power $2^2$
$\int2\sqrt{4\tan\left(\theta \right)^2+4}\sec\left(\theta \right)^2d\theta$
$\int2\sqrt{4}\sqrt{\tan\left(\theta \right)^2+1}\sec\left(\theta \right)^2d\theta$
Calculate the power $\sqrt{4}$
$\int2\cdot 2\sqrt{\tan\left(\theta \right)^2+1}\sec\left(\theta \right)^2d\theta$
When multiplying two powers that have the same base ($2$), you can add the exponents
$\int4\sqrt{\tan\left(\theta \right)^2+1}\sec\left(\theta \right)^2d\theta$
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The power of a product is equal to the product of it's factors raised to the same power
$\int4\sqrt{\tan\left(\theta \right)^2+1}\sec\left(\theta \right)^2d\theta$
Intermediate steps
Applying the trigonometric identity: $\tan(x)^2+1=\sec(x)^2$
$\int4\sec\left(\theta \right)\sec\left(\theta \right)^2d\theta$
When multiplying exponents with same base you can add the exponents
$\int4\sec\left(\theta \right)^{3}d\theta$
$\int4\sec\left(\theta \right)^{3}d\theta$
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The integral of a constant by a function is equal to the constant multiplied by the integral of the function
$4\int\sec\left(\theta \right)^{3}d\theta$
Intermediate steps
$4\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{\left(3-1\right)}}{3-1}+\frac{3-2}{3-1}\int\sec\left(\theta \right)^{\left(3-2\right)}d\theta\right)$
Subtract the values $3$ and $-1$
$4\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{3-1}+\frac{3-2}{3-1}\int\sec\left(\theta \right)^{\left(3-2\right)}d\theta\right)$
Add the values $3$ and $-1$
$4\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}+\frac{3-2}{3-1}\int\sec\left(\theta \right)^{\left(3-2\right)}d\theta\right)$
Subtract the values $3$ and $-2$
$4\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}+\frac{1}{3-1}\int\sec\left(\theta \right)^{\left(3-2\right)}d\theta\right)$
Add the values $3$ and $-1$
$4\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}+\frac{1}{2}\int\sec\left(\theta \right)^{\left(3-2\right)}d\theta\right)$
$4\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}+\frac{1}{2}\int\sec\left(\theta \right)^{\left(3-2\right)}d\theta\right)$
Subtract the values $3$ and $-2$
$4\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}+\frac{1}{2}\int\sec\left(\theta \right)^{1}d\theta\right)$
Any expression to the power of $1$ is equal to that same expression
$4\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}+\frac{1}{2}\int\sec\left(\theta \right)d\theta\right)$
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Simplify the integral $\int\sec\left(\theta \right)^{3}d\theta$ applying the reduction formula, $\displaystyle\int\sec(x)^{n}dx=\frac{\sin(x)\sec(x)^{n-1}}{n-1}+\frac{n-2}{n-1}\int\sec(x)^{n-2}dx$
$4\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}+\frac{1}{2}\int\sec\left(\theta \right)d\theta\right)$
Intermediate steps
$4\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}\right)+4\cdot \frac{1}{2}\int\sec\left(\theta \right)d\theta$
Multiply $4$ times $\frac{1}{2}$
$4\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}\right)+2\int\sec\left(\theta \right)d\theta$
Multiplying the fraction by $4$
$\frac{4\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}+2\int\sec\left(\theta \right)d\theta$
Take $\frac{4}{2}$ out of the fraction
$2\sin\left(\theta \right)\sec\left(\theta \right)^{2}+2\int\sec\left(\theta \right)d\theta$
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Solve the product $4\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}+\frac{1}{2}\int\sec\left(\theta \right)d\theta\right)$
$2\sin\left(\theta \right)\sec\left(\theta \right)^{2}+2\int\sec\left(\theta \right)d\theta$
Intermediate steps
The integral of the secant function is given by the following formula, $\displaystyle\int\sec(x)dx=\ln\left|\sec(x)+\tan(x)\right|$
$2\ln\left|\sec\left(\theta \right)+\tan\left(\theta \right)\right|$
Express the variable $\theta$ in terms of the original variable $x$
$2\ln\left|\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right|$
Add fraction's numerators with common denominators: $\frac{\sqrt{x^2+4}}{2}$ and $\frac{x}{2}$
$2\ln\left|\frac{\sqrt{x^2+4}+x}{2}\right|$
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The integral $2\int\sec\left(\theta \right)d\theta$ results in: $2\ln\left|\frac{\sqrt{x^2+4}+x}{2}\right|$
$2\ln\left|\frac{\sqrt{x^2+4}+x}{2}\right|$
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Gather the results of all integrals
$2\sin\left(\theta \right)\sec\left(\theta \right)^{2}+2\ln\left|\frac{\sqrt{x^2+4}+x}{2}\right|$
Intermediate steps
Multiplying the fraction by $2$
$\frac{2x}{\sqrt{x^2+4}}\frac{x^2+4}{4}+2\ln\left|\frac{\sqrt{x^2+4}+x}{2}\right|$
$\frac{2x\left(x^2+4\right)}{4\sqrt{x^2+4}}+2\ln\left|\frac{\sqrt{x^2+4}+x}{2}\right|$
Simplify the fraction by $x^2+4$
$\frac{2x\sqrt{x^2+4}}{4}+2\ln\left|\frac{\sqrt{x^2+4}+x}{2}\right|$
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Express the variable $\theta$ in terms of the original variable $x$
$\frac{2x\sqrt{x^2+4}}{4}+2\ln\left|\frac{\sqrt{x^2+4}+x}{2}\right|$
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Take $\frac{2}{4}$ out of the fraction
$\frac{1}{2}x\sqrt{x^2+4}+2\ln\left|\frac{\sqrt{x^2+4}+x}{2}\right|$
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As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration
$\frac{1}{2}x\sqrt{x^2+4}+2\ln\left|\frac{\sqrt{x^2+4}+x}{2}\right|+C_0$
$\frac{1}{2}x\sqrt{x^2+4}+2\ln\left|\frac{\sqrt{x^2+4}+x}{2}\right|+C_0$