Here, we show you a step-by-step solved example of integration by trigonometric substitution. This solution was automatically generated by our smart calculator:
We can solve the integral $\int\sqrt{x^2+4}dx$ by applying integration method of trigonometric substitution using the substitution
Differentiate both sides of the equation $x=2\tan\left(\theta \right)$
Find the derivative
The derivative of a function multiplied by a constant 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 $1$
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
The power of a product is equal to the product of it's factors raised to the same power
Simplify $4\tan\left(\theta \right)^2+4$ into secant function
The power of a product is equal to the product of it's factors raised to the same power
Multiply $2$ times $2$
When multiplying exponents with same base you can add the exponents: $4\sec\left(\theta \right)\sec\left(\theta \right)^2$
Substituting in the original integral, we get
The integral of a function times a constant ($4$) is equal to the constant times the integral of the function
Simplify the integral $\int\sec\left(\theta \right)^{3}d\theta$ applying the reduction formula, $\displaystyle\int\sec(x)^{n}dx=\frac{\sin(x)\sec(x)^{n-1}}{n-1}+\frac{n-2}{n-1}\int\sec(x)^{n-2}dx$
Solve the product $4\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}+\frac{1}{2}\int\sec\left(\theta \right)d\theta\right)$
Simplify the fraction $4\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}\right)$
The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$
Multiplying fractions $\frac{x}{\sqrt{x^2+4}} \times \frac{x^2+4}{4}$
Simplify the fraction by $x^2+4$
Multiplying the fraction by $2$
Express the variable $\theta$ in terms of the original variable $x$
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 $x$
The integral $2\int\sec\left(\theta \right)d\theta$ results in: $2\ln\left(\frac{\sqrt{x^2+4}+x}{2}\right)$
Gather the results of all integrals
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
Simplify the expression by applying the property of the logarithm of a quotient
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