Question
(1 point) Evaluate the indefinite integral. \( \int 5 e^{5 x} \sin \left(e^{5 x}\right) d x= \)
Ask by Lindsey Bowen. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The integral evaluates to \( -\cos(e^{5x}) + C \).
Solution
To evaluate the indefinite integral:
\[
\int 5 e^{5x} \sin \left(e^{5x}\right) dx
\]
Let's use substitution:
1. **Let** \( u = e^{5x} \).
2. **Then**, the derivative \( du = 5 e^{5x} dx \).
3. **Rewrite the integral** in terms of \( u \):
\[
\int \sin(u) \, du
\]
4. **Integrate**:
\[
-\cos(u) + C
\]
5. **Substitute back** \( u = e^{5x} \):
\[
-\cos(e^{5x}) + C
\]
**Final Answer:**
\[
-\cos\big(e^{5 x}\big) + C
\]
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Simplify this solution Extra Insights
To solve the indefinite integral \( \int 5 e^{5 x} \sin \left(e^{5 x}\right) d x \), we can use substitution. Let \( u = e^{5x} \), which means \( du = 5 e^{5x} dx \) or \( dx = \frac{du}{5 e^{5x}} = \frac{du}{u} \). Substituting these into the integral gives us: \[ \int \sin(u) \, du \] The integral of \( \sin(u) \) is \(-\cos(u) + C\). Thus, substituting back gives: \[ -\cos(e^{5x}) + C \] So, the solution to the integral is: \[ \int 5 e^{5 x} \sin \left(e^{5 x}\right) d x = -\cos \left(e^{5 x}\right) + C \]
