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Integral of 1a2 x2
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The term 1/(a^2 + x^2) is purely algebraic. However, its integral is trigonometric. When you look this up in a table of integrals, you'll find the integral: • ∫1/(a^2+x^2)dx = (1/a)arctan(x/a) + C • How is this so? • This is where Pythagora's Theorem and the rules of trigonometric come in to help. • We can't simply use a u-substitution to solve this problem. Instead, if we construct a right-angle triangle with the 2 shorter sides of lengths a and x, then the relationship between the hypotenuse and the the 2 shorter sides is a^2+x^2 (Pythagoras' Theorem). • Now tanθ = (x/a), or x = atanθ. • Taking the derivative with respect to θ, we get dx/dθ = sec^2θ. • Thus we need to substitute x = atanθ, and dx = sec^2θdθ to solve the integral. • Thanks for watching. Please give me a thumbs up if you have found this video helpful. • Please ask me a maths question by commenting below and I will try to help you in future videos. • Tip me some DogeCoin: A4f3URZSWDoJCkWhVttbR3RjGHRSuLpaP3 • Tip me at PayPal: https://paypal.me/MasterWu
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