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Table of Integrals

A table of integrals is extremely useful in technical mechanics as it provides a comprehensive collection of antiderivatives. By using this table, complex integration problems can be solved in less time as one does not have to integrate each individual function anew. This saves time and reduces errors.

The integral table allows engineers and technicians to navigate smoothly through calculations in mechanics, especially when working with frequently occurring functions. It provides a handy reference to expedite the integration process and ensure the accuracy of results.

The integral table is a valuable tool that enhances efficiency and increases the reliability of calculations in technical mechanics.

Table of Indefinite Integrals of Basic Functions

Table 1: Table of Integrals of Basic Functions
Function \(y=f(x)\) Antiderivative \(F(x)\)
Constant functions \(\int 0 ~ \mathrm{d}x = 0 \cdot \int \mathrm{d}x\) \(0 \cdot x + C = C\)
\(\int 1 ~ \mathrm{d}x = 1 \cdot \int \mathrm{d}x\)		 \(1 \cdot x + C = x + C\)
\(\int k ~ \mathrm{d}x = k \cdot \int \mathrm{d}x\) \(k \cdot x + C\)
Power functions		 \(\int x^n~\mathrm{d}x\) \(\dfrac{x^{n+1}}{n+1}+C\) für \(n \neq -1\)
\(\int x^{-1}~\mathrm{d}x = {\displaystyle \int} \dfrac{1}{x} ~\mathrm{d}x\) \(\ln|x|+C\) für \(x \neq 0\)
\(\int x ~\mathrm{d}x\)		 \(\dfrac{1}{2}x^2 + C\)
\(\int 2x ~\mathrm{d}x\) \(x^2 + C\)
\(\int x^2 ~\mathrm{d}x\) \(\dfrac{1}{3}x^3 + C\)
\(\int 3x^2 ~\mathrm{d}x\) \(x^3 + C\)
\({\displaystyle \int} -\dfrac{2}{x^3} ~\mathrm{d}x\) \(\dfrac{1}{x^2} + C\)
\({\displaystyle \int} -\dfrac{1}{x^2} ~\mathrm{d}x\) \(\dfrac{1}{x} + C\)
Square root functions \(\int \sqrt{x}~\mathrm{d}x\) \(\dfrac{2}{3} x^{\frac{3}{2}}+C\)
\(\int \sqrt[n]{x}~\mathrm{d}x\) \(\dfrac{n}{n+1} \left(\sqrt[n]{x}\right)^{n+1}+C\) für \(n \neq -1\)
\({\displaystyle \int} \dfrac{1}{\sqrt{x}}~\mathrm{d}x\) \(2 \sqrt{x}+C\)
\({\displaystyle \int} \dfrac{1}{n\left(\sqrt[n]{x^{n-1}}\right)}~\mathrm{d}x\) \(\sqrt[n]{x}+C\)
Exponential functions \(\int e^x ~\mathrm{d}x\) \(e^x+C\)
\(\int e^{\alpha x} ~\mathrm{d}x\) \(\dfrac{1}{\alpha}e^{\alpha x}+C\)
\(\int a^{x} ~\mathrm{d}x\) \(\dfrac{a^x}{\ln(a)}+C\) für \(a \in \mathbb{R}^+ / \{1\}\)
Logarithmic functions \(\int \ln(x) ~\mathrm{d}x\) \(x \cdot \ln(x)-x+C\)
\(\int \log_a x ~\mathrm{d}x\) \(\dfrac{1}{\ln(a)}(x \cdot \ln(x) - x)+C\)
\(\int a^{x} \cdot \ln(a) ~\mathrm{d}x\) \(a^x+C\) für \(a \in \mathbb{R}^+\)
Trigonometric functions \(\int \sin(x) ~\mathrm{d}x\) \(- \cos(x)+C\)
\(\int \cos(x) ~\mathrm{d}x\) \(\sin(x)+C\)
\({\displaystyle \int} \dfrac{1}{\sin^2(x)} ~\mathrm{d}x\) \(- \cot(x)+C\) \(x \neq k\pi\) mit \(k \in \mathbb{Z}\)
\({\displaystyle \int} \dfrac{1}{\cos^2(x)} ~\mathrm{d}x\) \(\tan(x)+C\) \(x \neq \dfrac{\pi}{2}+k\pi\) mit \(k \in \mathbb{Z}\)
Hyperbolic functions \(\int \sinh(x) ~\mathrm{d}x\) \(\cosh(x)+C\)
\(\int \cosh(x) ~\mathrm{d}x\) \(\sinh(x)+C\)
\({\displaystyle \int} \dfrac{1}{\sinh^2(x)} ~\mathrm{d}x\) \(- \coth(x)+C\) \(x \neq 0\)
\({\displaystyle \int} \dfrac{1}{\cosh^2(x)} ~\mathrm{d}x\) \(\tanh(x)+C\)

Table of Indefinite Integrals of Irrational Functions

Table 2: Table of Integrals of Irrational Functions
Function \(y=f(x)\) Antiderivative \(F(x)\)
\({\displaystyle \int} \sqrt{a^2-x^2} ~ \mathrm{d}x\)		 \(\dfrac{1}{2}\left(x \cdot \sqrt{a^2-x^2} + a^2 \cdot \arcsin\left(\dfrac{x}{a}\right) \right)\)
\({\displaystyle \int} x \sqrt{a^2-x^2} ~ \mathrm{d}x\)		 \(-\dfrac{1}{3} \left(\sqrt{a^2-x^2}\right)^3\)
\({\displaystyle \int} x^2 \sqrt{a^2-x^2} ~ \mathrm{d}x\)		 \(-\dfrac{x}{4}\left(\sqrt{a^2-x^2}\right)^3 + \dfrac{a^2}{8} \left(x \cdot \sqrt{a^2-x^2} + a^2 \cdot \arcsin\left(\dfrac{x}{a}\right) \right)\)
\({\displaystyle \int} \left(\sqrt{a^2-x^2}\right)^3 ~ \mathrm{d}x\)		 \(\dfrac{1}{4}\left(x \cdot \left(\sqrt{a^2-x^2}\right)^3 + \dfrac{3a^2x}{2} \cdot \sqrt{a^2-x^2}+\dfrac{3a^4}{2} \cdot \arcsin\left(\dfrac{x}{a}\right) \right)\)

Table of Indefinite Integrals of Trigonometric Functions

Table 3: Integral Table of Trigonometric Functions
Function \(y=f(x)\) Antiderivative \(F(x)\)
\({\displaystyle \int} \sin(ax) ~ \mathrm{d}x\)		 \(-\dfrac{1}{a}\cos(ax)+C\)
\({\displaystyle \int} \sin^2(ax) ~ \mathrm{d}x\)		 \(\dfrac{1}{2} x - \dfrac{1}{4a}\sin(2ax)+C\)
\({\displaystyle \int} \cos(ax) ~ \mathrm{d}x\)		 \(\dfrac{1}{a}\sin(ax)+C\)
\({\displaystyle \int} \cos^2(ax) ~ \mathrm{d}x\)		 \(\dfrac{1}{2} x + \dfrac{1}{4a}\sin(2ax)+C\)
\({\displaystyle \int} \sin(ax)\cos(ax) ~ \mathrm{d}x\)		 \(\dfrac{1}{2a}\sin^2(ax)+C\)
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