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Learn how to solve definite integrals problems step by step online.
$\int_{0}^{\frac{\pi}{3}}\tan\left(x\right)^5\sec\left(x\right)^4dx$
Learn how to solve definite integrals problems step by step online. Integrate the function tan(x)^5sec(x)^4 from 0 to pi/3. Simplifying. We identify that the integral has the form \int\tan^m(x)\sec^n(x)dx. If n is even, then the secant function is expressed as the tangent function. The factor \sec^n(x) is separated in two factors: \sec^2(x) and \left(\tan^2(x)+1\right)^{n-4}. We can solve the integral \int_{0}^{\frac{\pi}{3}}\tan\left(x\right)^5\left(\tan\left(x\right)^2+1\right)\sec\left(x\right)^2dx by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it u), which when substituted makes the integral easier. We see that \tan\left(x\right) it's a good candidate for substitution. Let's define a variable u and assign it to the choosen part. Now, in order to rewrite dx in terms of du, we need to find the derivative of u. We need to calculate du, we can do that by deriving the equation above.