Problem 24

Question

Use integration tables to evaluate the integral. $$ \int_{0}^{\pi} x \sin x d x $$

Step-by-Step Solution

Verified

Answer

The evaluated definite integral is $\pi$.

1Step 1: Identify the Method to Use

Recognize that the integral is a product of a linear function and a sine function, thus integration by parts will be used. The formula for integration by parts, found in integration tables, is $ \int u dv = uv - \int v du $.This formula will be applied with $u = x, dv = sin(x) dx$.

2Step 2: Compute the Derivative and the Integral

Calculate the derivative of u, $ du = dx $, and calculate the integral of $ dv, v = -cos(x) $.

3Step 3: Apply the Integration by Parts Formula

The integral becomes $ -x \cdot cos(x) - (- \int cos(x) dx) $. Evaluate the integral inside the parentheses, which gives $ sin(x) $, leading to $ -x \cdot cos(x) + sin(x) $.

4Step 4: Evaluate the Definite Integral

Substitute the limits of integration, from 0 to pi, to find the definite integral by calculating $-\pi cos(\pi) + sin(\pi) - (0 - 0)$ which simplifies to $-\pi \cdot -1 - 0 = \pi$.

Problem 24

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