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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
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$\int\left(\sec\left(x\right)^2-1\right)\sec\left(x\right)^3dx$
Learn how to solve trigonometric integrals problems step by step online. Solve the trigonometric integral int(tan(x)^2sec(x)^3)dx. 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)^3 by each term of the polynomial \left(\sec\left(x\right)^2-1\right). Expand the integral \int\left(\sec\left(x\right)^{5}-\sec\left(x\right)^3\right)dx into 2 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int\sec\left(x\right)^{5}dx results in: \frac{1}{4}\sec\left(x\right)^3\tan\left(x\right)+\frac{3\sec\left(x\right)\tan\left(x\right)}{8}+\frac{3}{8}\ln\left(\sec\left(x\right)+\tan\left(x\right)\right).