antiderivative

An antiderivative or indefinite integral of a function $f$ is a differentiable function $F$ whose derivative is $f$. Symbolically, $F^{\prime }=f$.

Antiderivatives can be used to compute the definite integral by the second part of the Fundamental Theorem of Calculus.

When an antiderivative $F$ of $f$ exists, then there are infinitely many antiderivatives for $f$, obtained by adding an arbitrary constant $C$ to $F$ called the constant of integration.

Common Antiderivatives

• $\int f^{\prime }(x)dx=f(x)+C$

• $\int 1dx=x+C$

• $\int adx=ax+C$

• $\int x^{n}dx=\frac{1}{n+1}{x}^{n+1}+C$
  except in the case $n=-1$:

• $\int \frac{1}{x}dx=\ln |x|+C$

• $\int e^{x}dx=e^x+C$

• $\int a^{x}dx=\frac{1}{\ln (a)}{a}^x+C$
  n.b. the logarithm is only defined for inputs $>0$

• $\int \sin (x)dx=-cos(x)+C$

• $\int \cos (x)dx=\sin (x)+C$

• $\int {\sec }^2(x)dx=\tan (x)+C$

• $\int {\csc }^2(x)dx=-\cot (x)+C$

• $\int \sec (x)\tan (x)dx=\sec (x)+C$

• $\int \csc (x)\cot (x)dx=-\csc (x)+C$

• $\int \frac{1}{\sqrt{a^2+x^2}}dx=\arcsin (\frac{x}{a})+C$

• $\int \frac{1}{a^2+x^2}dx=\frac{1}{a}\arctan (\frac{x}{a})+C$

See also
integration by parts