illustration of integration techniques

The following integralDlmfPlanetmath is an example that illustrates many integration techniques.

Problem. Determine the antiderivative of tanx.

. We start with substitution (

u =tanx
u2 =tanx
2udu =sec2xdx

Using the Pythagorean identity tan2x+1=sec2x, we obtain:

2udu =(tan2x+1)dx
2udu =(u4+1)dx
2uu4+1du =dx


tanx𝑑x =u2uu4+1𝑑u

For this last integral, we use the method of partial fractionsPlanetmathPlanetmath (

2u2(u2-u2+1)(u2+u2+1) =A+Buu2-u2+1+C+Duu2+u2+1
2u2 =(A+Bu)(u2+u2+1)+(C+Du)(u2-u2+1)

From this, we obtain the following system of equations:


This can be into two smaller systems of equations:


It is clear that the first system yields A=C=0, and it can easily be verified that B=12 and D=-12. Therefore,

tanx𝑑x =12uu2-u2+1𝑑u-12uu2+u2+1𝑑u

Now we make the following substitutions:


Note that we have v+12=u=w-12. Therefore,

tanx𝑑x =12v+12v2+12𝑑v-12w-12w2+12𝑑w

For the first and third integrals in the last expression, note that the numerator is a of the derivative of the denominator. For these, we use the formula


For the second and fourth integrals in the last expression, we use the formula


with a=12. Hence,

tanx𝑑x =122ln(v2+12)+12arctan(v2)-122ln(w2+12)+12arctan(w2)+K

(We use K for the constant of integration to avoid confusion with C from the system of equations.)

Title illustration of integration techniques
Canonical name IllustrationOfIntegrationTechniques
Date of creation 2013-03-22 17:50:16
Last modified on 2013-03-22 17:50:16
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 13
Author Wkbj79 (1863)
Entry type Example
Classification msc 26A36