Differentials Integrals

Reference data and engineering information about differentials integrals for miscellaneous applications.

differentialsintegrals

Overview

Common differentials and integrals used in engineering calculations.

#	Function	Integral
14	$x^n$	$\int x^n dx = \frac{x^{n+1}}{n+1}, \quad n \neq -1$
15	$\frac{1}{x}$	$\int \frac{1}{x} dx = \ln(x)$
16	$e^{ax}$	$\int e^{ax} dx = \frac{e^{ax}}{a}, \quad a \neq 0$
17	$a^x$	$\int a^x dx = \frac{a^x}{\ln(a)}, \quad a > 0, a \neq 1$
18	$\ln(x)$	$\int \ln(x) dx = x(\ln(x) - 1)$
19	$\sin(x)$	$\int \sin(x) dx = -\cos(x)$
20	$\cos(x)$	$\int \cos(x) dx = \sin(x)$
21	$\tan(x)$	$\int \tan(x) dx = -\ln(\cos(x))$
22	$\cot(x)$	$\int \cot(x) dx = \ln(\sin(x))$
23	$\sec^2(x)$	$\int \sec^2(x) dx = \tan(x)$
24	$\csc^2(x)$	$\int \csc^2(x) dx = -\cot(x)$
25	$\frac{1}{\sqrt{1-x^2}}$	$\int \frac{1}{\sqrt{1-x^2}} dx = \sin^{-1}(x), \quad \\|x\\| < 1$
26	$\frac{1}{1+x^2}$	$\int \frac{1}{1+x^2} dx = \tan^{-1}(x), \quad \\|x\\| < 1$

The following differential formulas are foundational. Some include important special cases:

Power Function (Special Case): When differentiating $x^x$ , the result is $\frac{d}{dx}x^x = x^x(1 + \ln x)$ .
Natural Logarithm: $\frac{d}{dx} \ln x = \frac{1}{x}$
Exponential Functions:
- $\frac{d}{dx} e^{ax} = a e^{ax}$
- $\frac{d}{dx} a^x = a^x \ln a$ (where $a > 0, a \neq 1$ )

The general form for integrating power functions has an important condition:

\int x^n dx = \frac{x^{n+1}}{n+1} + C \quad \text{for } n \neq -1

The integral corresponding to $n = -1$ is the natural logarithm. Note the standard absolute value inclusion in the result:

\int \frac{1}{x} dx = \ln |x| + C

Similarly, the integral for the natural logarithm itself is:

\int \ln x \, dx = x(\ln x - 1) + C