Explain why or why not. Determine whether the following statements are true and give an explanation or counterexample.
d. ∫2 sin x cos x dx = −(1/2) cos 2x + C.

Verified step by step guidance

1. Step 1: Recall the trigonometric identity for the product of sine and cosine: $2\; sin(x)\; cos(x)\; =\; sin(2x)$. This identity simplifies the integrand.
2. Step 2: Substitute the identity into the integral: $\int 2\; sin(x)\; cos(x)\; dx\; =\; \int sin(2x)\; dx$. This reduces the problem to finding the integral of $sin(2x)$.
3. Step 3: Recall the formula for the integral of $sin(kx)$: $\int sin(kx)\; dx\; =\; -(1/k)\; cos(kx)\; +\; C$. Here, $k\; =\; 2$.
4. Step 4: Apply the formula to compute the integral: $\int sin(2x)\; dx\; =\; -(1/2)\; cos(2x)\; +\; C$. This matches the given expression.
5. Step 5: Conclude that the statement is true because the integral calculation aligns with the given result: $\int 2\; sin(x)\; cos(x)\; dx\; =\; -(1/2)\; cos(2x)\; +\; C$.

Key Concepts

Integration of Trigonometric Functions

Integration of trigonometric functions often involves using identities to simplify the integrand. In this case, the identity sin(2x) = 2 sin(x) cos(x) can be applied to rewrite the integral, making it easier to evaluate. Understanding these identities is crucial for solving integrals involving products of sine and cosine.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links differentiation and integration, stating that if F is an antiderivative of f on an interval, then the integral of f from a to b is F(b) - F(a). This theorem is essential for confirming the correctness of an integral's evaluation and understanding the relationship between a function and its integral.

Constant of Integration

When performing indefinite integration, it is important to include a constant of integration (C) because the process of differentiation eliminates constant terms. This constant represents the family of antiderivatives and is crucial for expressing the general solution to an integral, ensuring that all possible functions that could yield the same derivative are accounted for.