Find the value of ∫_0^π sin(x) cos(x) dx.

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Question: Find the value of ∫_0^π sin(x) cos(x) dx.

Options:

Correct Answer: 0

Solution:

Using the identity sin(2x) = 2sin(x)cos(x), the integral becomes (1/2)∫_0^π sin(2x) dx = 0.

Find the value of ∫_0^π sin(x) cos(x) dx.

Find the value of ∫_0^π sin(x) cos(x) dx.

Step 1: Recognize the integral we need to solve: ∫_0^π sin(x) cos(x) dx.
Step 2: Use the trigonometric identity sin(2x) = 2sin(x)cos(x) to rewrite the integral.
Step 3: Rewrite sin(x)cos(x) as (1/2)sin(2x). So, the integral becomes ∫_0^π sin(x) cos(x) dx = (1/2)∫_0^π sin(2x) dx.
Step 4: Now, we need to calculate the integral ∫_0^π sin(2x) dx.
Step 5: Find the antiderivative of sin(2x), which is -1/2 cos(2x).
Step 6: Evaluate the definite integral from 0 to π: [-1/2 cos(2x)] from 0 to π.
Step 7: Calculate the value at the upper limit (π): -1/2 cos(2π) = -1/2 * 1 = -1/2.
Step 8: Calculate the value at the lower limit (0): -1/2 cos(0) = -1/2 * 1 = -1/2.
Step 9: Subtract the lower limit from the upper limit: (-1/2) - (-1/2) = 0.
Step 10: Multiply the result by (1/2) from Step 3: (1/2) * 0 = 0.

Trigonometric Identities – Understanding and applying the identity sin(2x) = 2sin(x)cos(x) to simplify integrals.
Definite Integrals – Evaluating definite integrals and understanding the properties of the sine function over specific intervals.