Problems on Techniques of Integration

We recognize here a rational function. The technique of integrationg such functions is the partial decomposition technique. One may also use the table for this integration since the degree of the denominator is 2.

The partial decomposition technique implies

$$\frac{3x-1}{x^2 + x -2} = \frac{3x-1}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x + 2}\;.$$

If we multiply both sides by $(x-1)(x+2)$, we get

$$A(x+2) + B(x-1) = 3x -1$$

which implies $A = \displaystyle \frac{2}{3}$ and $B = \displaystyle \frac{7}{3}$. So

$$\frac{3x-1}{x^2 + x -2} = \frac{1}{3} \left[\frac{2}{x-1} + \frac{7}{x + 2}\right]\;.$$

Hence

$$\int \frac{3x-1}{x^2 + x -2}dx = \frac{2}{3} \int \frac{1}{x-1}dx + \frac{7}{3}\int \frac{1}{x + 2}dx$$

which implies

$$\int \frac{x+1}{x^2 + 4}dx = \frac{2}{3} \ln\vert x-1\vert + \frac{7}{3} \ln\vert x+2\vert + C\;.$$

If you prefer to jump to the next problem, click on Next Problem below.

[Next Problem] [Matrix Algebra] [Trigonometry] [Calculus] [Geometry] [Algebra] [Differential Equations] [Complex Variables]

S.O.S MATHematics home page

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

Copyright © 1999-2024 MathMedics, LLC. All rights reserved.
Contact us
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA
users online during the last hour