Consider the integral ∫sinx cosx d x (a) Solve this integral by using u -substitution with u= si

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Consider the integral ∫sinx cosx d x (a) Solve this integral...

Consider the integral $\int \sin x \cos x d x$ (a) Solve this integral by using $u$ -substitution with $u=$ $\sin x$ and $d u=\cos x d x$ (b) Solve the integral another way, using $u$ -substitution with $u=\cos x$ and $d u=-\sin x d x$ (c) How must your two answers be related? Use algebra to prove this relationship.

Consider the integral $\int \sin x \cos x d x$
(a) Solve this integral by using $u$ -substitution with $u=$ $\sin x$ and $d u=\cos x d x$
(b) Solve the integral another way, using $u$ -substitution with $u=\cos x$ and $d u=-\sin x d x$
(c) How must your two answers be related? Use algebra to prove this relationship.

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Key Concepts

Indefinite Integrals and Antiderivative Equivalence

Indefinite integrals yield antiderivatives that are defined up to an arbitrary constant. When an integral is solved using different methods, the resulting antiderivatives may initially appear different; however, they are equivalent because any two antiderivatives of the same function differ only by a constant. Recognizing this equivalence is crucial for verifying the correctness of different integration techniques.

Algebraic Manipulation in Integration

Algebraic manipulation is used to transform and simplify expressions during the process of integration. It plays a key role in demonstrating that antiderivatives obtained through different substitution methods are consistent with each other, and in proving that they differ only by a constant. This underlines the importance of algebra in the validation of integral solutions.

U-Substitution

U-substitution is a fundamental technique in integration that simplifies an integrand by substituting a part of the expression with a new variable. This method relies on identifying a component of the integrand whose derivative is also present, thus transforming the integral into a form that is easier to integrate.

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