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Here, we show you a step-by-step solved example of integration by trigonometric substitution. This solution was automatically generated by our smart calculator:
$\int\sqrt{x^2+4}dx$
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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
Differentiate both sides of the equation $x=2\tan\left(\theta \right)$
$dx=\frac{d}{d\theta}\left(2\tan\left(\theta \right)\right)$
Find the derivative
$\frac{d}{d\theta}\left(2\tan\left(\theta \right)\right)$
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 ${f'(x) = sec^2(x)\cdot D_x(x)}$
$2\frac{d}{d\theta}\left(\theta \right)\sec\left(\theta \right)^2$
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 the polynomial $4\tan\left(\theta \right)^2+4$ by it's greatest common factor (GCF): $4$
$\int2\sqrt{4\left(\tan\left(\theta \right)^2+1\right)}\sec\left(\theta \right)^2d\theta$
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The power of a product is equal to the product of it's factors raised to the same power
$\int4\sqrt{\tan\left(\theta \right)^2+1}\sec\left(\theta \right)^2d\theta$
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Applying the trigonometric identity: $1+\tan\left(\theta \right)^2 = \sec\left(\theta \right)^2$
$\int4\sqrt{\sec\left(\theta \right)^2}\sec\left(\theta \right)^2d\theta$
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The integral of a function times a constant ($4$) is equal to the constant times the integral of the function
$4\int\sqrt{\sec\left(\theta \right)^2}\sec\left(\theta \right)^2d\theta$
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Simplify $\sqrt{\sec\left(\theta \right)^2}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $0.5$
$4\int\sec\left(\theta \right)\sec\left(\theta \right)^2d\theta$
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When multiplying exponents with same base you can add the exponents: $\sec\left(\theta \right)\sec\left(\theta \right)^2$
$4\int\sec\left(\theta \right)^{3}d\theta$
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Rewrite $\sec\left(\theta \right)^{3}$ as the product of two secants
$4\int\sec\left(\theta \right)^2\sec\left(\theta \right)d\theta$
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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
$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
Intermediate steps
Taking the derivative of secant function: $\frac{d}{dx}\left(\sec(x)\right)=\sec(x)\cdot\tan(x)\cdot D_x(x)$
$\frac{d}{d\theta}\left(\theta \right)\sec\left(\theta \right)\tan\left(\theta \right)$
The derivative of the linear function is equal to $1$
$\sec\left(\theta \right)\tan\left(\theta \right)$
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First, identify or choose $u$ and calculate it's derivative, $du$
$\begin{matrix}\displaystyle{u=\sec\left(\theta \right)}\\ \displaystyle{du=\sec\left(\theta \right)\tan\left(\theta \right)d\theta}\end{matrix}$
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Now, identify $dv$ and calculate $v$
$\begin{matrix}\displaystyle{dv=\sec\left(\theta \right)^2d\theta}\\ \displaystyle{\int dv=\int \sec\left(\theta \right)^2d\theta}\end{matrix}$
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Solve the integral to find $v$
$v=\int\sec\left(\theta \right)^2d\theta$
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The integral of $\sec(x)^2$ is $\tan(x)$
$\tan\left(\theta \right)$
Intermediate steps
When multiplying two powers that have the same base ($\tan\left(\theta \right)$), you can add the exponents
$4\left(\tan\left(\theta \right)\sec\left(\theta \right)-\int\sec\left(\theta \right)\tan\left(\theta \right)^2d\theta\right)$
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Now replace the values of $u$, $du$ and $v$ in the last formula
$4\left(\tan\left(\theta \right)\sec\left(\theta \right)-\int\sec\left(\theta \right)\tan\left(\theta \right)^2d\theta\right)$
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Multiply the single term $4$ by each term of the polynomial $\left(\tan\left(\theta \right)\sec\left(\theta \right)-\int\sec\left(\theta \right)\tan\left(\theta \right)^2d\theta\right)$
$4\tan\left(\theta \right)\sec\left(\theta \right)-4\int\sec\left(\theta \right)\tan\left(\theta \right)^2d\theta$
Intermediate steps
Applying the trigonometric identity: $\tan\left(\theta \right)^2 = \sec\left(\theta \right)^2-1$
$4\tan\left(\theta \right)\sec\left(\theta \right)-4\int\sec\left(\theta \right)\left(\sec\left(\theta \right)^2-1\right)d\theta$
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We identify that the integral has the form $\int\tan^m(x)\sec^n(x)dx$. If $n$ is odd and $m$ is even, then we need to express everything in terms of secant, expand and integrate each function separately
$4\tan\left(\theta \right)\sec\left(\theta \right)-4\int\sec\left(\theta \right)\left(\sec\left(\theta \right)^2-1\right)d\theta$
Intermediate steps
Multiply the single term $\sec\left(\theta \right)$ by each term of the polynomial $\left(\sec\left(\theta \right)^2-1\right)$
$\int\left(\sec\left(\theta \right)^2\sec\left(\theta \right)-\sec\left(\theta \right)\right)$
When multiplying exponents with same base you can add the exponents: $\sec\left(\theta \right)^2\sec\left(\theta \right)$
$4\tan\left(\theta \right)\sec\left(\theta \right)-4\int\left(\sec\left(\theta \right)^{3}-\sec\left(\theta \right)\right)d\theta$
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Multiply the single term $\sec\left(\theta \right)$ by each term of the polynomial $\left(\sec\left(\theta \right)^2-1\right)$
$4\tan\left(\theta \right)\sec\left(\theta \right)-4\int\left(\sec\left(\theta \right)^{3}-\sec\left(\theta \right)\right)d\theta$
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Expand the integral $\int\left(\sec\left(\theta \right)^{3}-\sec\left(\theta \right)\right)d\theta$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
$4\tan\left(\theta \right)\sec\left(\theta \right)-4\int\sec\left(\theta \right)^{3}d\theta-4\int-\sec\left(\theta \right)d\theta$
Intermediate steps
Express the variable $\theta$ in terms of the original variable $x$
$4\frac{x}{2}\frac{\sqrt{x^2+4}}{2}-4\int\sec\left(\theta \right)^{3}d\theta-4\int-\sec\left(\theta \right)d\theta$
Multiplying the fraction by $4\left(\frac{\sqrt{x^2+4}}{2}\right)$
$\sqrt{x^2+4}x-4\int\sec\left(\theta \right)^{3}d\theta-4\int-\sec\left(\theta \right)d\theta$
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Express the variable $\theta$ in terms of the original variable $x$
$\sqrt{x^2+4}x-4\int\sec\left(\theta \right)^{3}d\theta-4\int-\sec\left(\theta \right)d\theta$
Intermediate steps
Simplify the integral $\int\sec\left(\theta \right)^{3}d\theta$ applying the reduction formula, $\displaystyle\int\sec(x)^{n}dx=\frac{\sin(x)\sec(x)^{n-1}}{n-1}+\frac{n-2}{n-1}\int\sec(x)^{n-2}dx$
$-4\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{3-1}+\frac{3-2}{3-1}\int\sec\left(\theta \right)d\theta\right)$
Solve the product $-4\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{3-1}+\frac{3-2}{3-1}\int\sec\left(\theta \right)d\theta\right)$
$-4\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}\right)-2\int\sec\left(\theta \right)d\theta$
Simplify the fraction $-4\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}\right)$
$-2\sin\left(\theta \right)\sec\left(\theta \right)^{2}-2\int\sec\left(\theta \right)d\theta$
Express the variable $\theta$ in terms of the original variable $x$
$-\frac{1}{2}\sqrt{x^2+4}x-2\int\sec\left(\theta \right)d\theta$
The integral of the secant function is given by the following formula, $\displaystyle\int\sec(x)dx=\ln\left|\sec(x)+\tan(x)\right|$
$-\frac{1}{2}\sqrt{x^2+4}x-2\ln\left(\sec\left(\theta \right)+\tan\left(\theta \right)\right)$
Express the variable $\theta$ in terms of the original variable $x$
$-\frac{1}{2}\sqrt{x^2+4}x-2\ln\left(\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right)$
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The integral $-4\int\sec\left(\theta \right)^{3}d\theta$ results in: $-\frac{1}{2}\sqrt{x^2+4}x-2\ln\left(\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right)$
$-\frac{1}{2}\sqrt{x^2+4}x-2\ln\left(\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right)$
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Gather the results of all integrals
$\sqrt{x^2+4}x-2\ln\left(\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right)-\frac{1}{2}\sqrt{x^2+4}x-4\int-\sec\left(\theta \right)d\theta$
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Combining like terms $\sqrt{x^2+4}x$ and $-\frac{1}{2}\sqrt{x^2+4}x$
$\frac{1}{2}\sqrt{x^2+4}x-2\ln\left(\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right)-4\int-\sec\left(\theta \right)d\theta$
Intermediate steps
The integral of a function times a constant ($-1$) is equal to the constant times the integral of the function
$4\int\sec\left(\theta \right)d\theta$
The integral of the secant function is given by the following formula, $\displaystyle\int\sec(x)dx=\ln\left|\sec(x)+\tan(x)\right|$
$4\ln\left(\sec\left(\theta \right)+\tan\left(\theta \right)\right)$
Express the variable $\theta$ in terms of the original variable $x$
$4\ln\left(\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right)$
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The integral $-4\int-\sec\left(\theta \right)d\theta$ results in: $4\ln\left(\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right)$
$4\ln\left(\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right)$
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Gather the results of all integrals
$\frac{1}{2}\sqrt{x^2+4}x-2\ln\left(\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right)+4\ln\left(\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right)$
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Combining like terms $-2\ln\left(\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right)$ and $4\ln\left(\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right)$
$\frac{1}{2}\sqrt{x^2+4}x+2\ln\left(\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right)$
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The least common multiple (LCM) of a sum of algebraic fractions consists of the product of the common factors with the greatest exponent, and the uncommon factors
$L.C.M.=2$
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Combine and simplify all terms in the same fraction with common denominator $2$
$\frac{1}{2}\sqrt{x^2+4}x+2\ln\left(\frac{\sqrt{x^2+4}+x}{2}\right)$
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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 $C$
$\frac{1}{2}\sqrt{x^2+4}x+2\ln\left(\frac{\sqrt{x^2+4}+x}{2}\right)+C_0$
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Simplify the expression by applying logarithm properties
$\frac{1}{2}\sqrt{x^2+4}x+2\ln\left(\sqrt{x^2+4}+x\right)+C_1$
Final answer to the problem
$\frac{1}{2}\sqrt{x^2+4}x+2\ln\left(\sqrt{x^2+4}+x\right)+C_1$
