Question

find the most general antiderivative or indefinite integral
intleft(e^{-10 x}+11^{x}
ight) d x
intleft(e^{-10 x}+11^{x}
ight) d x=square

Explanation:

Step1: Integrate each term separately

Use the integral rules \(\int e^{ax}dx=\frac{1}{a}e^{ax}+C\) (\(a
eq0\)) and \(\int a^{x}dx=\frac{a^{x}}{\ln a}+C\) (\(a > 0,a
eq1\)).
For \(\int e^{-10x}dx\), let \(a=- 10\), then \(\int e^{-10x}dx=-\frac{1}{10}e^{-10x}+C_1\).
For \(\int11^{x}dx\), let \(a = 11\), then \(\int11^{x}dx=\frac{11^{x}}{\ln11}+C_2\).

Step2: Combine the results

\(\int(e^{-10x}+11^{x})dx=\int e^{-10x}dx+\int11^{x}dx\)
\(=-\frac{1}{10}e^{-10x}+\frac{11^{x}}{\ln11}+C\) (\(C = C_1 + C_2\))

Answer:

\(-\frac{1}{10}e^{-10x}+\frac{11^{x}}{\ln11}+C\)