Integration by Trigonometric Substitution Calculator

Differentiate both sides of the equation $x=2\tan\left(\theta \right)$

Find the derivative

The derivative of a function multiplied by a constant ($2$) is equal to the constant times the derivative of the function

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 ${f'(x) = sec^2(x)\cdot D_x(x)}$

The derivative of the linear function is equal to $1$

Calculate the power $2^2$

Calculate the power $\sqrt{4}$

Multiply $2$ times $2$

Applying the power of a power property

Subtract the values $3$ and $-2$

Any expression to the power of $1$ is equal to that same expression

Taking the derivative of secant function: $\frac{d}{dx}\left(\sec(x)\right)=\sec(x)\cdot\tan(x)\cdot D_x(x)$

When multiplying two powers that have the same base ($\tan\left(\theta \right)$), you can add the exponents

Multiply $4$ times $-1$

Apply the formula: $\int\sec\left(x\right)\tan\left(x\right)^2dx$$=\int\sec\left(x\right)^3dx-\int\sec\left(x\right)dx$, where $x=\theta $

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

Rewrite $\sec\left(\theta \right)^3$ as the product of two secants

We can solve the integral $\int\sec\left(\theta \right)^2\sec\left(\theta \right)d\theta$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula

First, identify $u$ and calculate $du$

Now, identify $dv$ and calculate $v$

Solve the integral

The integral of $\sec(x)^2$ is $\tan(x)$

Now replace the values of $u$, $du$ and $v$ in the last formula

Solve the product $-4\left(\tan\left(\theta \right)\sec\left(\theta \right)-\int\tan\left(\theta \right)^2\sec\left(\theta \right)d\theta\right)$

Multiply $-4$ times $-1$

Apply the formula: $\int\sec\left(x\right)\tan\left(x\right)^2dx$$=\int\sec\left(x\right)^3dx-\int\sec\left(x\right)dx$, where $x=\theta $

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

Cancel like terms $-4\ln\left(\sec\left(\theta \right)+\tan\left(\theta \right)\right)$ and $4\ln\left(\sec\left(\theta \right)+\tan\left(\theta \right)\right)$

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

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

Simplifying

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

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

Multiplying the fraction by $-2$

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

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