Integrals: Difference between revisions

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Revision as of 15:19, 24 May 2026

Integral Forms & Antiderivatives

Basic Forms

$\int u d v = u v - \int v d u$
$\int u^{n} d u = \frac{u^{n + 1}}{n + 1} + C, (n \neq - 1)$
$\int \frac{d u}{u} = \ln | u | + C$
$\int e^{u} d u = e^{u} + C$
$\int a^{u} d u = \frac{a^{u}}{\ln a} + C$
$\int \sin u d u = - \cos u + C$
$\int \cos u d u = \sin u + C$
$\int \sec^{2} u d u = \tan u + C$
$\int \csc^{2} u d u = - \cot u + C$
$\int \sec u \tan u d u = \sec u + C$
$\int \csc u \cot u d u = - \csc u + C$
$\int \tan u d u = \ln | \sec u | + C$
$\int \cot u d u = \ln | \sin u | + C$
$\int \sec u d u = \ln | \sec u + \tan u | + C$
$\int \csc u d u = \ln | \csc u - \cot u | + C$
$\int \frac{d u}{\sqrt{a^{2} - u^{2}}} = \sin^{- 1} \frac{u}{a} + C, (a > 0)$
$\int \frac{d u}{a^{2} + u^{2}} = \frac{1}{a} \tan^{- 1} \frac{u}{a} + C$
$\int \frac{d u}{u \sqrt{u^{2} - a^{2}}} = \frac{1}{a} \sec^{- 1} \frac{u}{a} + C$
$\int \frac{d u}{a^{2} - u^{2}} = \frac{1}{2 a} \ln | \frac{u + a}{u - a} | + C$
$\int \frac{d u}{u^{2} - a^{2}} = \frac{1}{2 a} \ln | \frac{u - a}{u + a} | + C$

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-Testing page creation and math.
-<math>E=mc^2</math>
+== Integral Forms & Antiderivatives ==
-<big>
+=== Basic Forms ===
-<big><math display="block">\int\frac{e^3/x}{x^2}\, dx</math></big></big>
+# <math>\int u \, dv = uv - \int v \, du</math><br>
+# <math>\int u^n \, du = \frac{u^{n+1}}{n+1} + C, \quad (n \neq -1)</math><br>
+# <math>\int \frac{du}{u} = \ln |u| + C</math><br>
+# <math>\int e^u \, du = e^u + C</math><br>
+# <math>\int a^u \, du = \frac{a^u}{\ln a} + C</math><br>
+# <math>\int \sin u \, du = -\cos u + C</math><br>
+# <math>\int \cos u \, du = \sin u + C</math><br>
+# <math>\int \sec^2 u \, du = \tan u + C</math><br>
+# <math>\int \csc^2 u \, du = -\cot u + C</math><br>
+# <math>\int \sec u \tan u \, du = \sec u + C</math><br>
+# <math>\int \csc u \cot u \, du = -\csc u + C</math><br>
+# <math>\int \tan u \, du = \ln |\sec u| + C</math><br>
+# <math>\int \cot u \, du = \ln |\sin u| + C</math><br>
+# <math>\int \sec u \, du = \ln |\sec u + \tan u| + C</math><br>
+# <math>\int \csc u \, du = \ln |\csc u - \cot u| + C</math><br>
+# <math>\int \frac{du}{\sqrt{a^2 - u^2}} = \sin^{-1} \frac{u}{a} + C, \quad (a > 0)</math><br>
+# <math>\int \frac{du}{a^2 + u^2} = \frac{1}{a} \tan^{-1} \frac{u}{a} + C</math><br>
+# <math>\int \frac{du}{u\sqrt{u^2 - a^2}} = \frac{1}{a} \sec^{-1} \frac{u}{a} + C</math><br>
+# <math>\int \frac{du}{a^2 - u^2} = \frac{1}{2a} \ln \left| \frac{u+a}{u-a} \right| + C</math><br>
+# <math>\int \frac{du}{u^2 - a^2} = \frac{1}{2a} \ln \left| \frac{u-a}{u+a} \right| + C</math><br>

Integrals: Difference between revisions

Revision as of 15:19, 24 May 2026

Integral Forms & Antiderivatives

Basic Forms

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