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Integration by trigonometric substitution Calculator

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1

Solved example of integration by trigonometric substitution

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

We can solve the integral $\int\sqrt{x^2+4}dx$ by applying integration method of trigonometric substitution using the substitution

$x=2\tan\left(\theta \right)$

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 $1$

$2\sec\left(\theta \right)^2$
3

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

$dx=2\sec\left(\theta \right)^2d\theta$
4

Substituting in the original integral, we get

$\int2\sqrt{4\tan\left(\theta \right)^2+4}\sec\left(\theta \right)^2d\theta$
5

Factor by the greatest common divisor $4$

$\int2\sqrt{4\left(\tan\left(\theta \right)^2+1\right)}\sec\left(\theta \right)^2d\theta$

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

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

$\int4\sqrt{\tan\left(\theta \right)^2+1}\sec\left(\theta \right)^2d\theta$
6

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

$\int4\sqrt{\tan\left(\theta \right)^2+1}\sec\left(\theta \right)^2d\theta$

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

$\int4\sec\left(\theta \right)^{3}d\theta$
7

Simplifying

$\int4\sec\left(\theta \right)^{3}d\theta$
8

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

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

$4\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{\left(3-1\right)}}{3-1}+\frac{3-2}{3-1}\int\sec\left(\theta \right)^{\left(3-2\right)}d\theta\right)$

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 $-1$

$4\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}+\frac{1}{2}\int\sec\left(\theta \right)^{\left(3-2\right)}d\theta\right)$

Divide $1$ by $2$

$4\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}+\frac{1}{2}\int\sec\left(\theta \right)^{\left(3-2\right)}d\theta\right)$

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

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

Simplify the integral $\int\sec\left(\theta \right)^{3}d\theta$ applying the reduction formula, $\displaystyle\int\sec(x)^{n}dx=\frac{\sin(x)\sec(x)^{n-1}}{n-1}+\frac{n-2}{n-1}\int\sec(x)^{n-2}dx$

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

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

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

$2\sin\left(\theta \right)\sec\left(\theta \right)^{2}+2\int\sec\left(\theta \right)d\theta$
10

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

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

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 $\frac{x}{2}$

$2\ln\left|\frac{\sqrt{x^2+4}+x}{2}\right|$
11

The integral $2\int\sec\left(\theta \right)d\theta$ results in: $2\ln\left|\frac{\sqrt{x^2+4}+x}{2}\right|$

$2\ln\left|\frac{\sqrt{x^2+4}+x}{2}\right|$
12

Gather the results of all integrals

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

Multiplying the fraction by $2$

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

Multiplying fractions

$\frac{2x\left(x^2+4\right)}{4\sqrt{x^2+4}}+2\ln\left|\frac{\sqrt{x^2+4}+x}{2}\right|$

Simplify the fraction by $x^2+4$

$\frac{2x\sqrt{x^2+4}}{4}+2\ln\left|\frac{\sqrt{x^2+4}+x}{2}\right|$
13

Express the variable $\theta$ in terms of the original variable $x$

$\frac{2x\sqrt{x^2+4}}{4}+2\ln\left|\frac{\sqrt{x^2+4}+x}{2}\right|$
14

Take $\frac{2}{4}$ out of the fraction

$\frac{1}{2}x\sqrt{x^2+4}+2\ln\left|\frac{\sqrt{x^2+4}+x}{2}\right|$
15

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

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

Final Answer

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

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