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The difficulty with integrating products (and likewise quotients) arises from the fact that differentiation is a derivation. The Jacobi identity / Leibniz rule / product rule rules this world and not the chain rule.WWGD said:On its own, just as a trick, ##sinxcosx=\frac{sin2x}{2}##, with simple integral ##\frac{-Cos2x}{2}##
But, yes, that denominator kills it. Maybe Fresh can write an insight on integrating expressions a/b from the respective integrals of a,b , right, Fresh? ;)
When integrating sin(x) or cos(x), you can use the trigonometric identities to simplify the expression. For sin(x), you can use the identity ∫sin(x) dx = -cos(x) + C and for cos(x), you can use ∫cos(x) dx = sin(x) + C, where C is the constant of integration.
The integral of sec^2(x) dx is tan(x) + C, where C is the constant of integration. Similarly, the integral of csc^2(x) dx is -cot(x) + C.
When integrating products of trigonometric functions, you can use integration by parts or trigonometric identities to simplify the expression. It often involves using trigonometric substitutions or recognizing patterns to integrate the product.
Integrating tan(x) involves using the substitution method or trigonometric identities to simplify the expression. The integral of tan(x) dx is -ln|cos(x)| + C. Similarly, the integral of cot(x) dx is ln|sin(x)| + C.
When integrating powers of trigonometric functions, you can use trigonometric identities or substitution methods to simplify the expression. It often involves applying the power rule or recognizing patterns to integrate the powers of trigonometric functions.