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Integrals of Rational Functions of Sine and Cosine Calculator

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Example
Created on 2024-06-20Asked by Mason Davis (Solvelet student)
Evaluate the integral ∫sin⁡xcos⁡2x dx \int \frac{\sin x}{\cos^2 x} \, dx .

Solution

To evaluate the integral ∫sin⁡xcos⁡2x dx \int \frac{\sin x}{\cos^2 x} \, dx : Use trigonometric identities to rewrite the integrand: sin⁡xcos⁡2x=sin⁡x1−sin⁡2x \frac{\sin x}{\cos^2 x} = \frac{\sin x}{1 - \sin^2 x} Now, let u=sin⁡x u = \sin x , then du=cos⁡x dx du = \cos x \, dx : ∫sin⁡x1−sin⁡2x dx=∫11−u2 du \int \frac{\sin x}{1 - \sin^2 x} \, dx = \int \frac{1}{1 - u^2} \, du This is a standard integral: ∫11−u2 du=12ln⁡∣1+u1−u∣+C \int \frac{1}{1 - u^2} \, du = \frac{1}{2} \ln \left| \frac{1 + u}{1 - u} \right| + C Substitute u=sin⁡x u = \sin x : 12ln⁡∣1+sin⁡x1−sin⁡x∣+C \frac{1}{2} \ln \left| \frac{1 + \sin x}{1 - \sin x} \right| + C Therefore, the integral ∫sin⁡xcos⁡2x dx \int \frac{\sin x}{\cos^2 x} \, dx is: ∫sin⁡xcos⁡2x dx=12ln⁡∣1+sin⁡x1−sin⁡x∣+C \int \frac{\sin x}{\cos^2 x} \, dx = \frac{1}{2} \ln \left| \frac{1 + \sin x}{1 - \sin x} \right| + C Where C C is the constant of integration. Solved on Solvelet with Basic AI Model
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DefinitionIntegrals of Rationals of Sine and Cosine involve integrals of polynomials where a factor of sine or cosine appears as a numerator of a denominator. Solving these integrals frequently requires using trigonometric identies, substitutions and partial fraction decomposition. They are frequently encountered in trigonometry, physics, and engineering. Applications: Then the textbook show a list of integrals that based on the integral ∫cos(x)·sin(u) dx can leads to arctan(cos(x))+C.Find and apply the integral ∫d(u)/[u^2+1] which intuition behind it come from tangent-derivative-chain rule-mulitplication-rule-velocity-and-acceleration, yield either sum-factor-quotient rule fundamentally based on the clear approach to antiderivative by axioms in rapid calculus written-since-thy-users can verify my work. Improper Integral Discussion (10 problems): 2quizzes and 0learning results Practice Improper Integral Discussion (5 grads) Instant trainer=siaosx Shaped like a camel Instantaneous Solution, Sine Function, High Cross Verification, Crossed Elimination (15 Billion Classical Ideas)(143 Separated Idea Total 99%)
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