Question
Determine the following indefinite integral. \( \int \frac{1}{15 y} d y \)
Ask by Barber Daniel. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The integral of \( \frac{1}{15 y} \) with respect to \( y \) is \( \frac{1}{15} \ln |y| + C \), where \( C \) is the constant of integration.
Solution
To evaluate the indefinite integral:
\[
\int \frac{1}{15 y} \, dy
\]
**Step 1: Factor Out the Constant**
The constant \(\frac{1}{15}\) can be factored out of the integral:
\[
\frac{1}{15} \int \frac{1}{y} \, dy
\]
**Step 2: Integrate \(\frac{1}{y}\)**
Recall that the integral of \(\frac{1}{y}\) with respect to \(y\) is the natural logarithm of the absolute value of \(y\):
\[
\int \frac{1}{y} \, dy = \ln |y| + C
\]
where \(C\) is the constant of integration.
**Step 3: Combine the Results**
Multiply the result by the factored-out constant:
\[
\frac{1}{15} \ln |y| + C
\]
**Final Answer:**
\[
\int \frac{1}{15 y} \, dy = \frac{1}{15} \ln |y| + C
\]
where \(C\) is the constant of integration.
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Simplify this solution The Deep Dive
To evaluate the integral \( \int \frac{1}{15 y} d y \), we can factor out the constant \( \frac{1}{15} \): \[ \int \frac{1}{15 y} d y = \frac{1}{15} \int \frac{1}{y} d y \] The integral of \( \frac{1}{y} \) is \( \ln |y| + C \), where \( C \) is the constant of integration. Thus, we have: \[ \frac{1}{15} \int \frac{1}{y} d y = \frac{1}{15} \ln |y| + C \] So, the final answer is: \[ \int \frac{1}{15 y} d y = \frac{1}{15} \ln |y| + C \]
