Here, we show you a step-by-step solved example of intégration cyclique par parties. This solution was automatically generated by our smart calculator:
Apply the formula: $\int\sec\left(\theta \right)^ndx$$=\int\sec\left(\theta \right)^2\sec\left(\theta \right)^{\left(n-2\right)}dx$, where $n=3$
We can solve the integral $\int\sec\left(x\right)^2\sec\left(x\right)dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula
Apply the trigonometric identity: $\frac{d}{dx}\left(\sec\left(\theta \right)\right)$$=\frac{d}{dx}\left(\theta \right)\sec\left(\theta \right)\tan\left(\theta \right)$
Apply the formula: $\frac{d}{dx}\left(x\right)$$=1$
First, identify or choose $u$ and calculate it's derivative, $du$
Now, identify $dv$ and calculate $v$
Solve the integral to find $v$
Apply the formula: $\int\sec\left(\theta \right)^2dx$$=\tan\left(\theta \right)+C$
Apply the formula: $x\cdot x$$=x^2$, where $x=\tan\left(x\right)$
Now replace the values of $u$, $du$ and $v$ in the last formula
Applying the trigonometric identity: $\tan\left(\theta \right)^2 = \sec\left(\theta \right)^2-1$
We identify that the integral has the form $\int\tan^m(x)\sec^n(x)dx$. If $n$ is odd and $m$ is even, then we need to express everything in terms of secant, expand and integrate each function separately
Multiply the single term $\sec\left(x\right)$ by each term of the polynomial $\left(\sec\left(x\right)^2-1\right)$
Apply the formula: $x\cdot x^n$$=x^{\left(n+1\right)}$, where $x^nx=\sec\left(x\right)^2\sec\left(x\right)$, $x=\sec\left(x\right)$, $x^n=\sec\left(x\right)^2$ and $n=2$
Multiply the single term $\sec\left(x\right)$ by each term of the polynomial $\left(\sec\left(x\right)^2-1\right)$
Expand the integral $\int\left(\sec\left(x\right)^{3}-\sec\left(x\right)\right)dx$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
Apply the formula: $1x$$=x$, where $x=\int\sec\left(x\right)dx$
Simplify the expression
Apply the formula: $\int\sec\left(\theta \right)dx$$=\ln\left(\sec\left(\theta \right)+\tan\left(\theta \right)\right)+C$
The integral $\int\sec\left(x\right)dx$ results in: $\ln\left(\sec\left(x\right)+\tan\left(x\right)\right)$
This integral by parts turned out to be a cyclic one (the integral that we are calculating appeared again in the right side of the equation). We can pass it to the left side of the equation with opposite sign
Moving the cyclic integral to the left side of the equation
Adding the integrals
Move the constant term $2$ dividing to the other side of the equation
The integral results in
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 $C$
Multiply the single term $\frac{1}{2}$ by each term of the polynomial $\left(\tan\left(x\right)\sec\left(x\right)+\ln\left(\sec\left(x\right)+\tan\left(x\right)\right)\right)$
Expand and simplify
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