integral of sqrt{[1-sqrt(x)] / [1+sqrt(x)]}

integral of sqrt{[1-sqrt(x)] / [1+sqrt(x)]}

use the substitution sqrt(x) = cost

use trigonometric formulae to simplify [1-cos t]  and [ 1+ cos t], sint using the half angle formulae

cancel off the common factors

simplify then again use trigonometric formulae to change the square terms to first degree expressions before integrating.

again use trigonometric formulae to change the variable back to x

trigonometric identities 

formulae on integration
 
PAGE 1 BASIC INTEGRATION

PAGE 2 INTEGRATION BY SUBSTITUTION

 PAGE 3 INTEGRATION BY COMPLETION OF SQUARES

PAGE 4 INTEGRATION BY PARTS

PAGE 5 INTEGRATION BY MANIPULATION OF NUMERATOR IN TERMS OF DENOMINATOR


PAGE 6 INTEGRATION USING PARTIAL FRACTIONS

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integral of 1 / [cos(x+a)cos(x+b)]

integral of 1 / [cos(x+a)cos(x+b)] by manipulating the angle in the numerator in terms of the angles of the denominator

introduce a  term sin(a-b) in the numerator by multiplying the numerator and denominator with it.

Then express the angle  (a-b)  in terms of the denominator(x+a) and (x+b)
using (a-b) = (x+a) - (x+b)

use trigonometric formula for expanding the numerator and then split the numerator

then simplify and integrate the two terms separately.

PAGE 1 BASIC INTEGRATION

PAGE 2 INTEGRATION BY SUBSTITUTION

 PAGE 3 INTEGRATION BY COMPLETION OF SQUARES

PAGE 4 INTEGRATION BY PARTS

PAGE 5 INTEGRATION BY MANIPULATION OF NUMERATOR IN TERMS OF DENOMINATOR


PAGE 6 INTEGRATION USING PARTIAL FRACTIONS

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integral of 1 / sqrt[4 + 3x - x² ] using completion of squares method

integral of 1 / sqrt[4 + 3x - x² ] using completion of squares method

integrate after completing the square
to make the coefficient of x²  (+1) , take -1 common , but it remains inside the sqrt sign
to get sqrt [ - ( x² -3x - 4)] now take half the coefficient of x , that is half of (-3)
to get [x- (3/2) ]² and then adjust the constant term and finally re introduce the -1 taken  out earlier

PAGE 1 BASIC INTEGRATION

PAGE 2 INTEGRATION BY SUBSTITUTION 

PAGE 3 INTEGRATION BY COMPLETION OF SQUARES

PAGE 4 INTEGRATION BY PARTS

PAGE 5 INTEGRATION BY MANIPULATION OF NUMERATOR IN TERMS OF DENOMINATOR

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integral using completion of squares method

integral using completion of squares method

integral of  1 / [(x^2) - x +1 ]

completion of squares method is used if there is a (x^2) term and a x term
  remember to make the coefficient of x² unity
take half the coefficient of x  that is half of  (-1)
so that you get [ x -(1/2) ]² , then adjust the constant term

PAGE 1 BASIC INTEGRATION
PAGE 2 INTEGRATION BY SUBSTITUTIONPAGE 3 INTEGRATION BY COMPLETION OF SQUARES
PAGE 4 INTEGRATION BY PARTS
PAGE 5 INTEGRATION BY MANIPULATION OF NUMERATOR IN TERMS OF DENOMINATOR




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some integrals

A solution to the integral of sqrt(tan (x) ) is available at http://rbmix.com/math/sqrtan/sqrtan.php

A solution to the integral of ln(sin(x)) with limits 0 to pi/2 using properties of definite integrals is available at http://rbmix.com/problem/int/int.php

A solution of the integral of [e^(ax) ] *cos(bx) using the technique of integration by parts is available at http://rbmix.com/problem/bypart/bpt.php

A solution to the integral of sec³(x) using the method of integration by parts is available at http://rbmix.com/problem/i2/i2.php

In the last two problems , the question is taken as I and then integration by parts is applied, and then the original integral I is manipulated out of the resultant integral and then extracted.

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index of math problems


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