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Differentials Integrals

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

differentialsintegrals

Overview

Common differentials and integrals used in engineering calculations.

Differentials

# Function Derivative
1 xnx^n ddxxn=nxn−1\frac{d}{dx} x^n = n x^{n-1}
2 ln⁡(x)\ln(x) ddxln⁡(x)=1x\frac{d}{dx} \ln(x) = \frac{1}{x}
3 eaxe^{ax} ddxeax=aeax\frac{d}{dx} e^{ax} = a e^{ax}
4 axa^x ddxax=axln⁡(a)\frac{d}{dx} a^x = a^x \ln(a)
5 xxx^x ddxxx=xx(1+ln⁡(x))\frac{d}{dx} x^x = x^x (1 + \ln(x))
6 sin⁡(x)\sin(x) ddxsin⁡(x)=cos⁡(x)\frac{d}{dx} \sin(x) = \cos(x)
7 cos⁡(x)\cos(x) ddxcos⁡(x)=−sin⁡(x)\frac{d}{dx} \cos(x) = -\sin(x)
8 tan⁡(x)\tan(x) ddxtan⁡(x)=sec⁡2(x)\frac{d}{dx} \tan(x) = \sec^2(x)
9 cot⁡(x)\cot(x) ddxcot⁡(x)=−csc⁡2(x)\frac{d}{dx} \cot(x) = -\csc^2(x)
10 sin⁡−1(x)\sin^{-1}(x) ddxsin⁡−1(x)=11−x2\frac{d}{dx} \sin^{-1}(x) = \frac{1}{\sqrt{1-x^2}}
11 cos⁡−1(x)\cos^{-1}(x) ddxcos⁡−1(x)=−11−x2\frac{d}{dx} \cos^{-1}(x) = \frac{-1}{\sqrt{1-x^2}}
12 tan⁡−1(x)\tan^{-1}(x) ddxtan⁡−1(x)=11+x2\frac{d}{dx} \tan^{-1}(x) = \frac{1}{1+x^2}
13 cot⁡−1(x)\cot^{-1}(x) ddxcot⁡−1(x)=−11+x2\frac{d}{dx} \cot^{-1}(x) = \frac{-1}{1+x^2}

Integrals

# Function Integral
14 xnx^n ∫xndx=xn+1n+1,n≠−1\int x^n dx = \frac{x^{n+1}}{n+1}, \quad n \neq -1
15 1x\frac{1}{x} ∫1xdx=ln⁡(x)\int \frac{1}{x} dx = \ln(x)
16 eaxe^{ax} ∫eaxdx=eaxa,a≠0\int e^{ax} dx = \frac{e^{ax}}{a}, \quad a \neq 0
17 axa^x ∫axdx=axln⁡(a),a>0,a≠1\int a^x dx = \frac{a^x}{\ln(a)}, \quad a > 0, a \neq 1
18 ln⁡(x)\ln(x) ∫ln⁡(x)dx=x(ln⁡(x)−1)\int \ln(x) dx = x(\ln(x) - 1)
19 sin⁡(x)\sin(x) ∫sin⁡(x)dx=−cos⁡(x)\int \sin(x) dx = -\cos(x)
20 cos⁡(x)\cos(x) ∫cos⁡(x)dx=sin⁡(x)\int \cos(x) dx = \sin(x)
21 tan⁡(x)\tan(x) ∫tan⁡(x)dx=−ln⁡(cos⁡(x))\int \tan(x) dx = -\ln(\cos(x))
22 cot⁡(x)\cot(x) ∫cot⁡(x)dx=ln⁡(sin⁡(x))\int \cot(x) dx = \ln(\sin(x))
23 sec⁡2(x)\sec^2(x) ∫sec⁡2(x)dx=tan⁡(x)\int \sec^2(x) dx = \tan(x)
24 csc⁡2(x)\csc^2(x) ∫csc⁡2(x)dx=−cot⁡(x)\int \csc^2(x) dx = -\cot(x)
25 11−x2\frac{1}{\sqrt{1-x^2}} ∫11−x2dx=sin⁡−1(x),∥x∥<1\int \frac{1}{\sqrt{1-x^2}} dx = \sin^{-1}(x), \quad \|x\| < 1
26 11+x2\frac{1}{1+x^2} ∫11+x2dx=tan⁡−1(x),∥x∥<1\int \frac{1}{1+x^2} dx = \tan^{-1}(x), \quad \|x\| < 1

Extended Formulas

Differential Formulas with Notes

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

  • Power Function (Special Case): When differentiating xxx^x, the result is ddxxx=xx(1+ln⁡x)\frac{d}{dx}x^x = x^x(1 + \ln x).
  • Natural Logarithm: ddxln⁡x=1x\frac{d}{dx} \ln x = \frac{1}{x}
  • Exponential Functions:
    • ddxeax=aeax\frac{d}{dx} e^{ax} = a e^{ax}
    • ddxax=axln⁡a\frac{d}{dx} a^x = a^x \ln a (where a>0,a≠1a > 0, a \neq 1)

Integral Formulas with Conditions

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

∫xndx=xn+1n+1+Cfor n≠−1\int x^n dx = \frac{x^{n+1}}{n+1} + C \quad \text{for } n \neq -1

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

∫1xdx=ln⁡∣x∣+C\int \frac{1}{x} dx = \ln |x| + C

Similarly, the integral for the natural logarithm itself is:

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

References