Engineering Handbook/Mathematics/Integral

Indefinite Integral

 * $$\int f(x)dx$$


 * {| class="wikitable" width=90% style="text-align:center; margin: 1em auto 1em auto"

!colspan=3 style="font-size:1.2em"|Table of Properties of Integrals !width=20%| !width=50%|Rule !Conditions !colspan=3| !style="background:#ebf0f7; font-size:1.2em"| Notes:
 * height=55| 1
 * $$\int a\,dx = ax$$
 * height=55| 2 Homogeniety
 * $$\int af(x) \,dx = a\int f(x)\,dx$$
 * height=55| 3 Associativity
 * $$ \int{ \left( f \pm g \pm h \pm \cdots \right) \,dx} = \int f \,dx \pm \int g \,dx \pm \int h \,dx \pm \cdots$$
 * height=55| 4 Integration by Parts
 * $$\int_a^b f g'\,dx = \left[ f g \right]_{a}^{b} - \int_a^b g f' \,dx$$
 * height=55| 4 General Integration by Parts
 * $$\int f^{(n)} g \,dx = f^{(n-1)}g' - f^{(n-2)}g'' + \ldots + (-1)^n \int f g^{(n)} \,dx$$
 * height=55| 5
 * $$\int f(ax) \,dx = \frac{1}{a} \int f(x) \, dx$$
 * height=55| 6 Substitution Rule
 * $$\int g \{ f (x) \} \,dx = \int g(u) \frac{dx}{du} \, du = \int \frac{g(u)}{f'(x) } \,du$$
 * $$u= f(x)\,$$
 * height=55| 7
 * $$\int x^n \,dx = \frac{x^{n+1}}{n+1}$$
 * $$n \ne -1\,$$
 * height=55| 8
 * $$\int \frac{1}{x} \,dx = \ln |x|$$
 * height=55| 9
 * $$\int e^x \, dx = e^x$$
 * height=55| 10
 * $$\int a^x \,dx = \frac{a^x}{\ln |a|}$$
 * $$a \ne 1$$
 * height=55| 6 Substitution Rule
 * $$\int g \{ f (x) \} \,dx = \int g(u) \frac{dx}{du} \, du = \int \frac{g(u)}{f'(x) } \,du$$
 * $$u= f(x)\,$$
 * height=55| 7
 * $$\int x^n \,dx = \frac{x^{n+1}}{n+1}$$
 * $$n \ne -1\,$$
 * height=55| 8
 * $$\int \frac{1}{x} \,dx = \ln |x|$$
 * height=55| 9
 * $$\int e^x \, dx = e^x$$
 * height=55| 10
 * $$\int a^x \,dx = \frac{a^x}{\ln |a|}$$
 * $$a \ne 1$$
 * $$\int e^x \, dx = e^x$$
 * height=55| 10
 * $$\int a^x \,dx = \frac{a^x}{\ln |a|}$$
 * $$a \ne 1$$
 * $$\int a^x \,dx = \frac{a^x}{\ln |a|}$$
 * $$a \ne 1$$
 * colspan=2 style="text-align:left"|
 * 1) f, g, h are functions of x
 * 2) a, n are constants.
 * 3) The constant of integration, C has been omitted from this table. It should be included in the working of the equation if applicable.
 * colspan = 13 style="text-align:right"|
 * }
 * }


 * {| class="wikitable" width=90% style="text-align:center; margin: 1em auto 1em auto"

!width=6%| !width=32%|Integral !width=32%|Value !width=22%|Remarks
 * 1
 * $$\int c \,dx$$
 * $$cx+C\,$$||
 * 2
 * $$\int x^n\,dx$$
 * $$\frac{x^{n+1}}{n+1}+C$$
 * $$n \ne -1$$
 * 3
 * $$\int \frac{1}{x}\,dx$$
 * $$\ln{\left|x\right|}+C$$||
 * 4
 * $$\int {1 \over {a^2+x^2}} \,dx$$
 * $${1 \over a}\arctan {x \over a} + C$$||
 * 5
 * $$\int {1 \over \sqrt{a^2-x^2}} \,dx$$
 * $$\arcsin {x \over a} + C$$||
 * 6
 * $$\int {-1 \over \sqrt{a^2-x^2}} \,dx$$
 * $$\arccos {x \over a} + C$$||
 * 7
 * $$\int {1 \over x\sqrt{x^2-a^2}} \,dx$$
 * $${1 \over a}\mbox{arcsec}\,{|x| \over a} + C$$||
 * 8
 * $$\int \ln {x}\,dx$$
 * $$x \ln {x} - x + C \,$$||
 * 9
 * $$\int \log_b {x}\,dx$$
 * $$x\log_b {x} - x\log_b {e} + C\,$$||
 * 10
 * $$\int e^x\,dx$$
 * $$e^x + C \,$$||
 * 11
 * $$\int a^x\,dx$$
 * $$\frac{a^x}{\ln{a}} + C$$||
 * 12
 * $$\int \sin{x}\, dx$$
 * $$-\cos{x} + C\,$$||
 * 13
 * $$\int \cos{x}\, dx $$
 * $$\sin{x} + C\,$$||
 * 14
 * $$\int \tan{x} \, dx$$
 * $$-\ln{\left| \cos {x} \right|} + C\,$$||
 * 15
 * $$\int \cot{x} \, dx$$
 * $$\ln{\left| \sin{x} \right|} + C\,$$||
 * 16
 * $$\int \sec{x} \, dx$$
 * $$\ln{\left| \sec{x} + \tan{x}\right|} + C\,$$||
 * 17
 * $$\int \csc{x} \, dx $$
 * $$-\ln{\left| \csc{x} + \cot{x}\right|} + C\,$$||
 * 18
 * $$\int \sec^2 x \, dx $$
 * $$\tan x + C\,$$||
 * 19
 * $$\int \csc^2 x \, dx$$
 * $$-\cot x + C\,$$||
 * 20
 * $$\int \sec{x} \, \tan{x} \, dx$$
 * $$\sec{x} + C\,$$||
 * 21
 * $$\int \csc{x} \, \cot{x} \, dx$$
 * $$- \csc{x} + C\,$$||
 * 22
 * $$\int \sin^2 x \, dx$$
 * $$\frac{1}{2}(x - \sin x \cos x) + C\,$$||
 * 23
 * $$\int \cos^2 x \, dx$$
 * $$\frac{1}{2}(x + \sin x \cos x) + C\,$$||
 * 24
 * $$\int \sin^n x \, dx$$
 * $$- \frac{\sin^{n-1} {x} \cos {x}}{n} + \frac{n-1}{n} \int \sin^{n-2}{x} \, dx$$||
 * 25
 * $$\int \cos^n x \, dx$$
 * $$- \frac{\cos^{n-1} {x} \sin {x}}{n} + \frac{n-1}{n} \int \cos^{n-2}{x} \, dx$$||
 * 26
 * $$\int \arctan{x} \, dx$$
 * $$x \, \arctan{x} - \frac{1}{2} \ln{\left| 1 + x^2\right|} + C$$||
 * 27
 * $$\int \sinh x \, dx$$
 * $$\cosh x + C\,$$||
 * 28
 * $$\int \cosh x \, dx $$
 * $$\sinh x + C\,$$||
 * 29
 * $$\int \tanh x \, dx$$
 * $$\ln |\cosh x| + C\,$$||
 * 30
 * $$\int \mbox{csch}\,x \, dx$$
 * $$\ln\left| \tanh {x \over2}\right| + C$$||
 * 31
 * $$\int \mbox{sech}\,x \, dx$$
 * $$\arctan(\sinh x) + C\,$$||
 * 32
 * $$\int \coth x \, dx$$
 * $$\ln|\sinh x| + C\,$$||
 * }
 * 25
 * $$\int \cos^n x \, dx$$
 * $$- \frac{\cos^{n-1} {x} \sin {x}}{n} + \frac{n-1}{n} \int \cos^{n-2}{x} \, dx$$||
 * 26
 * $$\int \arctan{x} \, dx$$
 * $$x \, \arctan{x} - \frac{1}{2} \ln{\left| 1 + x^2\right|} + C$$||
 * 27
 * $$\int \sinh x \, dx$$
 * $$\cosh x + C\,$$||
 * 28
 * $$\int \cosh x \, dx $$
 * $$\sinh x + C\,$$||
 * 29
 * $$\int \tanh x \, dx$$
 * $$\ln |\cosh x| + C\,$$||
 * 30
 * $$\int \mbox{csch}\,x \, dx$$
 * $$\ln\left| \tanh {x \over2}\right| + C$$||
 * 31
 * $$\int \mbox{sech}\,x \, dx$$
 * $$\arctan(\sinh x) + C\,$$||
 * 32
 * $$\int \coth x \, dx$$
 * $$\ln|\sinh x| + C\,$$||
 * }
 * $$\int \mbox{sech}\,x \, dx$$
 * $$\arctan(\sinh x) + C\,$$||
 * 32
 * $$\int \coth x \, dx$$
 * $$\ln|\sinh x| + C\,$$||
 * }
 * }

Definite Integral

 * $$\int_a^b f(x), dx$$
 * $$\int_{a}^{b} f(x)\,dx = F(b) - F(a).$$

Furthermore, for every x in the interval (a, b),


 * $$\frac{d}{dx}\int_a^x f(t)\, dt = f(x).$$