Mathematical formulas

Calculus II

Subjects Home Search Guide

Sections

1Notation, domains and conventions
2Indefinite integral and properties
3Definite integral and Fundamental Theorem of Calculus
4Substitution method
5Integration by parts
6Trigonometric integrals
7Trigonometric substitution and quadratic radicals
8Rational functions and partial fractions
9Improper integrals
10Applications of the integral
11Numerical integration
12Parametric curves
13Polar coordinates
14Sequences and series
15Power series and Taylor series
16Elementary differential equations
17Guide to choosing a method
AAppendix A: extensive table of antiderivatives
BAppendix B: useful equivalences

Search formulas…

Language

Loading formula…

Mathematical formulas · API-served content · Guide

About Reviews Contact

Home
Calculus II
Typical reduction formulas
Formula

5.3. Typical reduction formulas

Formula

In 5.3. Typical reduction formulas

Formula

\int x^n e^{ax}\,dx =\frac{x^n e^{ax}}{a}-\frac{n}{a}\int x^{n-1}e^{ax}\,dx, \qquad a\neq0

Tags

antiderivative
integration by parts: reduction formulas
by parts
reduction
tecnica fundamental

Variables

$x$ Variable independiente (variable de integración)
$a$ Constante real (a menudo $a>0$ )
$n$ Exponente o índice entero
$dx$ Diferencial respecto de $x$
$a\neq0$ Base of the natural logarithm

Conditions

$a\neq0$

Found an error? Contact us.

Related formulas

Related
It comes from the product rule:
$\int u\,dv=uv-\int v\,du.$
Open formula Open in section
Related
Formula
$\int_a^b u\,dv=[uv]_a^b-\int_a^b v\,du.$
Open formula Open in section
Related
Formula
$\int \sin^n x\,dx =-\frac{\sin^{n-1}x\cos x}{n} +\frac{n-1}{n}\int\sin^{n-2}x\,dx, \qquad n\ge2$
Open formula Open in section
Related
Formula
$\int \cos^n x\,dx =\frac{\cos^{n-1}x\sin x}{n} +\frac{n-1}{n}\int\cos^{n-2}x\,dx, \qquad n\ge2$
Open formula Open in section
Related
Formula
$\int \tan^n x\,dx =\frac{\tan^{n-1}x}{n-1}-\int\tan^{n-2}x\,dx, \qquad n>1$
Open formula Open in section
Related
Formula
$\int \cot^n x\,dx =-\frac{\cot^{n-1}x}{n-1}-\int\cot^{n-2}x\,dx, \qquad n>1$
Open formula Open in section
Related
Formula
$\int \sec^n x\,dx =\frac{\sec^{n-2}x\tan x}{n-1} +\frac{n-2}{n-1}\int\sec^{n-2}x\,dx, \qquad n>1$
Open formula Open in section
Related
Formula
$\int \csc^n x\,dx =-\frac{\csc^{n-2}x\cot x}{n-1} +\frac{n-2}{n-1}\int\csc^{n-2}x\,dx, \qquad n>1$
Open formula Open in section