overall accuracy: 85.7% record: 5 score: 5 ∫e^(-5x)dx = 5e^(-5x)+c -5e^(-5x)+c e^(-5x)+c -1/5e^(-5x)+c

Explanation:

Step1: Use substitution method

Let $u = - 5x$, then $du=-5dx$, and $dx=-\frac{1}{5}du$.

Step2: Substitute into integral

$\int e^{-5x}dx=\int e^{u}\cdot(-\frac{1}{5})du=-\frac{1}{5}\int e^{u}du$.

Step3: Integrate $e^{u}$

Since $\int e^{u}du = e^{u}+C$, we have $-\frac{1}{5}\int e^{u}du=-\frac{1}{5}e^{u}+C$.

Step4: Substitute back $u = - 5x$

$-\frac{1}{5}e^{u}+C=-\frac{1}{5}e^{-5x}+C$.

Answer:

$-\frac{1}{5}e^{-5x}+C$