integral using substitution and then integration by parts

integral using substitution and then integration by parts

integral of [1 / (x^4)][sqrt( 1 + (x^2))][ log( 1 + (x^2)) - 2log( x)]

first simplify using property of logarithms

take x^2 common from the sqrt term obtain [1+1/(x^2)] and try to get the same term inside the log expression

cancel off the x to get 1 /[x^3]   then use substitution

then use integration by parts with log(t) as the first function

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 / [a + bsinx + c cosx ]

Integral of 1 / [a + bsinx + c cosx ]

Integration formulae

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you might then have to restort to completion of squares in some cases after the manipulation

Examples
* integral of 1 / (5+4cosx) with limits 0 to (pi/2)
integral of 1 / {5 + 4 cos x } with limits




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integration using partial fractions

Integration by partial fractions

Integration formulae

first of all check if degree of numerator is less than degree of denominator
else perform long division.

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you might then have to restore to completion of squares in some cases after the manipulation
in some cases the manipulation can be done through inspection

Examples

* ∫ dx / [ (x-2) (1 + x²)]
integral using partial fractions

*integral of (x² +1) / [(x+3)(x-1)²]
integration using partial fraction







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manipulation of numerator in terms of the denominator

Integration by manipulation of numerator in terms of the denominator

Integration formulae

(px+q) / (ax² + bx +c) , (px+q) /sqrt(ax² + bx +c) ,

(px² + qx +r) /(ax² + bx +c) type

(psinx+qcosx ) / (a sinx + b cosx )

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for

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you might then have to restore to completion of squares in some cases after the manipulation
in some cases the manipulation can be done through inspection

Examples

* ∫ { (4x - 7) / (x² + x +1)} dx
integration of linear expression in numerator mixed with quadratic expression

* ∫ { cosx/ [ sinx +cosx] } dx
integral of [ cos x ] / [sin x + cos x ]

*not the above types but still an example of manupulating the angle in the numerator in terms of the angles in the denominator


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

*integral of { (cos x)^2 / [ (cos x)^2+ 4 (sin x)^2 ]  }
answer and a little explanation of integral of { (cos x)^2 / [ (cos x)^2+ 4 (sin x)^2 ]  }



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integration of some products using by parts

Integration by parts

Integration formulae

If an integral is a product of the form uv

usually u is chosen so that successive derivatives of u will become zero after a finite number of steps
provided v has a nice integral or you can make the choice using ILATE rule
inverse , log, algebraic, trigonometric, exponential

for eg. for integral of [x² cosx]

u = x² , v = cosx

but for integral of [x² ln(x)] , you should make it into integral of [ ln(x) * x²]
so that u = ln(x) , v = x²

for integral of lnx, make it into integral of [lnx * 1] with u = lnx , v = 1

in some integrals, like integral of [ (e^x) cosx ], take the integral as I, apply integration by parts a couple of times or so, and then obtain the original integral I again, then rearrange and extract the value of I.

also note the type
here split into two integrals
use integration by parts once on integral of [f(x)exp(x)] taking f(x) as the first function
this will cancel off the integral of [f '(x)exp(x)] leaving you with exp(x) * f(x ) + C which is the required answer

Examples on integration by parts


* integral of ln(x)
answer and some explanation


*integral of arc(cosx)
answer and some explanation


* integral of { [ arcsin(sqrt(x)) - arccos(sqrt(x)) ] / [ arcsin(sqrt(x)) + arccos(sqrt(x))] }
explanation and answer to the integral using integration by parts is available here

*integral of [1 / (x^4)][sqrt( 1 + (x^2))][ log( 1 + (x^2)) - 2log( x)]
explanation and answer using substitution and then integration by parts is available here

*integral of  [e^2x]cos3x using integration by parts
explanation and answer of   integral of [e^2x]cos3x using integration by parts


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integration by completion of squares

Integration by completion of squares

Integration formulae

If an integral is of

sqrt(ax² + bx +c) , 1 /sqrt(ax² + bx +c) ,1 /(ax² + bx +c) type

first complete the square

ax² + bx +c = a[ x² + (b/a)x + (c/a) ] =a[ ( x +(b/2a) )² + (c/a) -(b²/4a²) ]

remember to make the coefficient of x² unity before starting
Examples on integration by completion of squares

* integral of [ 1 / (x² -x+1)]
integral using completion of squares method

*integral of 1 / sqrt[4 + 3x - x² ]
completion of squares inside a square root

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integration substitution

substitution method for integration

Integration formulae

If an integral is not of a standard type, but can be reduced to a standard type by substitution

of the form
u = f(x)
du = f ' (x) dx take the differential on both sides if necessary

after substitution, remember that everything should be in terms of u alone

Examples on integration by substitution


* integral of x^3 / sqrt(9 - x^2)
answer and some explanation

* integral of (x^3)*sqrt[4-x^2]
answer and some explanation


* integral of 1 / { x*sqrt[ax-x^2]} using substitution
explanation of the answer to integration by substitution

*integral of 1 / 6{ [ x^(1/2)+ x^(1/3) ] } using substitution
answer with some explanation 


*integral of sqrt{[1-sqrt(x)] / [1+sqrt(x)]}
explanation and answer to the problem on integration

*integral of 2*cube of (tanx) with 0 to pi/4 as limits
explanation and answer to integral of 2*cube of (tanx) with 0 to pi/4 as limits




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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