Integrals of Familiar Functions
Constant Rule
The integral of a constant \(a\) is \(ax+C\).
\[\int a \, dx= ax +C\]
Power Rules
For \(n \neq -1\),
\[\int x^n \, dx= \frac{1}{n+1} x^{n+1} +C\]
and
\[
\int \frac{1}{x} \, dx = \ln (x) + C
\]
Trig Rules
\[\int \sin(x) \, dx = -\cos(x) +C\]
and
\[\int \cos(x) \, dx= \sin(x) + C\]
Exponential Rules
\[\int e^x \, dx=e^x +C\]
and for any positive number \(a >0\),
\[\int a^x \, dx =\frac{1}{\ln(a)}a^x + C\]
Logarithmic Rules
\[\int \ln (x) \, dx =x \ln(x) -x + C\]
and for any positive number \(a >0\),
\[\int \log_a (x) \, dx = x \log_a(x) - \frac{1}{\ln (a)} x + C \]
Arithmetic Rules for Integrals
Constant Multiple Rule
If \(c\) is a constant, then
\[\int cf(x) \, dx =c \int f(x) \, dx\]
Sum and Difference Rules
\[\int (f(x)+g(x)) \,dx= \int f(x) \, dx + \int g(x)\, dx\]
and
\[\int (f(x)-g(x)) \,dx= \int f(x) \, dx - \int g(x)\, dx\]
Substitution Rule
For any constant \(k\),
\[\int f'(kx) = \frac{1}{k} f(kx) + C\]