Mcdonald Bryant
06/15/2023 · High School
Evaluate \( \int e^{x} \sin x d x \)
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Tutor-Verified Answer
Step-by-step Solution
To evaluate the integral \( \int e^{x} \sin x \, dx \), we can use the method of integration by parts or the technique of integration by parts twice. However, a more systematic approach is to use the formula for integrating products of exponentials and trigonometric functions.
We will use the formula:
\[
\int e^{ax} \sin(bx) \, dx = \frac{e^{ax}}{a^2 + b^2} (a \sin(bx) - b \cos(bx)) + C
\]
In our case, we have \( a = 1 \) and \( b = 1 \). Thus, we can substitute these values into the formula:
\[
\int e^{x} \sin x \, dx = \frac{e^{x}}{1^2 + 1^2} (1 \sin x - 1 \cos x) + C
\]
Calculating \( 1^2 + 1^2 \):
\[
1^2 + 1^2 = 1 + 1 = 2
\]
Now substituting this back into the formula gives:
\[
\int e^{x} \sin x \, dx = \frac{e^{x}}{2} (\sin x - \cos x) + C
\]
Thus, the final result is:
\[
\int e^{x} \sin x \, dx = \frac{e^{x}}{2} (\sin x - \cos x) + C
\]
Quick Answer
\[
\int e^{x} \sin x \, dx = \frac{e^{x}}{2} (\sin x - \cos x) + C
\]
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