QuestionCalculate the integral: $\int \frac{2 b}{b^{2}+7} d b$

Studdy Solution

STEP 1

Assumptions1. The integral is a definite integral. We are integrating with respect to $b$
3. The integral is of the form $\int \frac{f'(x)}{f(x)} dx$ , which is the integral of a derivative divided by the function itself. This is a standard form that can be solved with a logarithmic function.

STEP 2

Identify the function $f(b)$ and its derivative $f'(b)$ in the integral. Here, $f(b) = b^2 +7$ and $f'(b) =2b$ .

STEP 3

Use the formula for the integral of a derivative divided by the function itself, which is $\int \frac{f'(x)}{f(x)} dx = \log |f(x)| + C$ , where $C$ is the constant of integration.

STEP 4

Substitute $f(b)$ and $f'(b)$ into the formula to solve the integral.
$\int \frac{2b}{b^2 +7} db = \log |b^2 +7| + C$ This is the solution to the integral.

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