Questions: ∫ from -1 to 0 of x/(x²+1) dx

Transcript text: $\int_{-1}^{0} \frac{x}{x^{2}+1} dx$

Solution

Solution Steps

Step 1: Identify the Integral

The problem asks us to evaluate the definite integral $\int_{-1}^{0} \frac{x}{x^{2}+1} \, dx$.

Step 2: Choose a Substitution

To solve this integral, we use the substitution method. Let $ u = x^2 + 1 $.

Step 3: Compute the Differential

Find the differential $ du $ by differentiating $ u $ with respect to $ x $: \[ \frac{du}{dx} = 2x \quad \Rightarrow \quad du = 2x \, dx \] This implies $ x \, dx = \frac{1}{2} du $.

Step 4: Change the Limits of Integration

Change the limits of integration according to the substitution:

When $ x = -1 $, $ u = (-1)^2 + 1 = 2 $.
When $ x = 0 $, $ u = (0)^2 + 1 = 1 $.

Step 5: Substitute and Simplify the Integral

Substitute into the integral: \[ \int_{-1}^{0} \frac{x}{x^{2}+1} \, dx = \int_{2}^{1} \frac{1}{u} \cdot \frac{1}{2} \, du = \frac{1}{2} \int_{2}^{1} \frac{1}{u} \, du \]

Step 6: Integrate

Integrate $\frac{1}{u}$ with respect to $ u $: \[ \frac{1}{2} \left[ \ln|u| \right]_{2}^{1} = \frac{1}{2} (\ln|1| - \ln|2|) \]

Step 7: Evaluate the Expression

Since $\ln|1| = \ln(1) = 0$, the expression simplifies to: \[ \frac{1}{2} (0 - \ln(2)) = -\frac{1}{2} \ln(2) \]