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Here, we show you a step-by-step solved example of cyclic integration by parts. This solution was automatically generated by our smart calculator:
$\int sec^3\left(x\right)dx$
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Rewrite $\sec\left(x\right)^3$ as the product of two secants
$\int\sec\left(x\right)^2\sec\left(x\right)dx$
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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
$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
Zwischenschritte
Taking the derivative of secant function: $\frac{d}{dx}\left(\sec(x)\right)=\sec(x)\cdot\tan(x)\cdot D_x(x)$
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
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
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
Multiply the single term $\frac{1}{2}$ by each term of the polynomial $\left(\tan\left(x\right)\sec\left(x\right)+\ln\left(\sec\left(x\right)+\tan\left(x\right)\right)\right)$