Integration by Trigonometric Substitution Calculator

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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:

We can solve the integral $\int\sqrtdx$ by applying integration method of trigonometric substitution using the substitution

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

Intermediate steps

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

$dx=\frac\left(2\tan\left(\theta \right)\right)$

Find the derivative

$\frac\left(2\tan\left(\theta \right)\right)$

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$2\frac\left(\tan\left(\theta \right)\right)$

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 $$, then $$

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

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

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

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

$dx=2\sec\left(\theta \right)^2d\theta$ Explain this step further

Intermediate steps

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

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

Simplify $4\tan\left(\theta \right)^2+4$ into secant function

$\int2\sqrt<4\sec\left(\theta \right)^2>\sec\left(\theta \right)^2d\theta$

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

$\int2\cdot 2\sec\left(\theta \right)\sec\left(\theta \right)^2d\theta$

Multiply $2$ times $2$

$\int4\sec\left(\theta \right)\sec\left(\theta \right)^2d\theta$

When multiplying exponents with same base you can add the exponents: $4\sec\left(\theta \right)\sec\left(\theta \right)^2$

$\int4\sec\left(\theta \right)^<3>d\theta$

Substituting in the original integral, we get

$\int4\sec\left(\theta \right)^<3>d\theta$ Explain this step further

The integral of a function times a constant ($4$) is equal to the constant times the integral of the function

$4\int\sec\left(\theta \right)^<3>d\theta$

Intermediate steps

Simplify the integral $\int\sec\left(\theta \right)^d\theta$ applying the reduction formula, $\displaystyle\int\sec(x)^dx=\frac>+\frac\int\sec(x)^dx$

$4\left(\frac<\sin\left(\theta \right)\sec\left(\theta \right)^>+\frac\int\sec\left(\theta \right)^d\theta\right)$

Subtract the values $3$ and $-1$

$4\left(\frac<\sin\left(\theta \right)\sec\left(\theta \right)^<2>>+\frac\int\sec\left(\theta \right)^d\theta\right)$

Subtract the values $3$ and $-1$

$4\left(\frac<\sin\left(\theta \right)\sec\left(\theta \right)^>+\frac\int\sec\left(\theta \right)^d\theta\right)$

Subtract the values $3$ and $-1$

$4\left(\frac<\sin\left(\theta \right)\sec\left(\theta \right)^>+\frac\int\sec\left(\theta \right)^d\theta\right)$

Subtract the values $3$ and $-2$

$4\left(\frac<\sin\left(\theta \right)\sec\left(\theta \right)^>+\frac\int\sec\left(\theta \right)^d\theta\right)$

Subtract the values $3$ and $-2$

$4\left(\frac<\sin\left(\theta \right)\sec\left(\theta \right)^>+\frac\int\sec\left(\theta \right)^d\theta\right)$

Simplify the integral $\int\sec\left(\theta \right)^d\theta$ applying the reduction formula, $\displaystyle\int\sec(x)^dx=\frac>+\frac\int\sec(x)^dx$

$4\left(\frac<\sin\left(\theta \right)\sec\left(\theta \right)^>+\frac\int\sec\left(\theta \right)^d\theta\right)$ Explain this step further

Intermediate steps

$4\left(\frac<\sin\left(\theta \right)\sec\left(\theta \right)^>\right)+4\cdot \left(\frac\right)\int\sec\left(\theta \right)^d\theta$

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

$4\left(\frac<\sin\left(\theta \right)\sec\left(\theta \right)^>\right)+4\cdot \left(\frac\right)\int\sec\left(\theta \right)d\theta$

Multiply the fraction and term in $4\cdot \left(\frac\right)\int\sec\left(\theta \right)d\theta$

$4\left(\frac<\sin\left(\theta \right)\sec\left(\theta \right)^>\right)+\frac\int\sec\left(\theta \right)d\theta$

Any expression multiplied by $1$ is equal to itself