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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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$\int\left(\sec\left(x\right)^2-1\right)\sec\left(x\right)dx$
Learn how to solve trigonometric integrals problems step by step online. Solve the trigonometric integral int(tan(x)^2sec(x))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) 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. The integral \int\sec\left(x\right)^{3}dx results in: \tan\left(x\right)\sec\left(x\right)-\int\sec\left(x\right)^3dx+\ln\left(\sec\left(x\right)+\tan\left(x\right)\right).