Constant Rule:
\( \int k \, dx = k x + C \)
Constant Multiple Rule:
\( \int k \cdot f(x) \, dx = k \int f(x) \, dx\)
Power Rule:
\( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{for } n \neq -1 \)
\( \int (ax+b)^n \, dx = \frac{(ax+b)^{n+1}}{a(n+1)} + C \)
Exponential Functions:
\( \int e^x \, dx = e^x + C \)
\( \int e^{ax} \, dx = \frac{e^{ax}}{a} + C \)
\( \int xe^x \, dx = e^x(x-1) + C \)
\( \int a^x \, dx = \frac{a^x}{\ln a} + C \)
Logarithmic Function:
\( \int \frac{1}{x} \, dx = \ln|x| + C \)
\( \int \ln x \, dx = x(\ln x-1) + C \)
\( \int \frac{1}{ax+b} \, dx = \frac{\ln(ax+b)}{a} + C\)
Trigonometric Functions:
\( \int \sin(x) \, dx = -\cos(x) + C \)
\( \int \cos(x) \, dx = \sin(x) + C \)
\( \int \sec^2(x) \, dx = \tan(x) + C \)
\( \int \csc^2(x) \, dx = -\cot(x) + C \)
\( \int \sec(x) \cdot \tan(x) \, dx = \sec(x) + C \)
\( \int \csc(x) \cdot \cot(x) \, dx = -\csc(x) + C \)
\( \int \tan(x) \, dx = \ln|\sec(x)| + C \)
\( \int \sec(x) \, dx = \ln|\sec(x) + \tan(x)| + C \)
\( \int \sin(ax+b) \, dx = \frac{-\cos(ax+b)}{a} + C \)
Sum/Diff Rule:
\( \int \left[ f(x) \pm g(x) \right] \, dx = \int f(x) \, dx \pm \int g(x) \, dx \)
Product Rule:
\( \int \left[ u \cdot v \right] \, dx = u\int v \, dx – \int[\frac{du}{dx} \times \int v \, dx] \, dx \)
Integration by Parts:
\( \int u \, dv = uv – \int v \, du \)
Substitution Rule:
\( \int f(g(x)) g'(x) \, dx = \int f(u) \, du \quad \text{where } u = g(x) \)
Inverse Trigonometric Functions:
\( \int \frac{1}{\sqrt{1-x^2}} \, dx = \arcsin(x) + C \)
\( \int \frac{-1}{\sqrt{1-x^2}} \, dx = \arccos(x) + C \)
\( \int \frac{1}{1+x^2} \, dx = \arctan(x) + C \)
\( \int \frac{-1}{1+x^2} \, dx = \text{arccot}(x) + C \)
Hyperbolic Functions:
\( \int \sinh(x) \, dx = \cosh(x) + C \)
\( \int \cosh(x) \, dx = \sinh(x) + C \)
\( \int \text{sech}^2(x) \, dx = \tanh(x) + C \)