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Solved Examples on Indefinite Integral

Illustration 1:Integrate the curve x/ (1+x4). (1978)

Solution:Let I = ∫xdx / (1 + x4)

= 1/2.∫ 2x / (1 + (x2)2) dx

Put x2 = u then 2xdx = du

Hence, I = ∫ du/ 2(1 + u2)

              = 1/2 tan-1 u +c

              = 1/2 tan-1(x2)+c

Illustration 2: Integrate sin x. sin 2x.sin 3x + sec2x. cos2 2x + sin4 x cos4 x. (1979)

Solution:Let I1 = ∫sin x. sin 2x. sin 3x dx

= 1/4 ∫ sin 4x + sin 2x - sin 6x)dx

= -cos 4x/16 -cos 2x/8 + cos 6x/24

I2 = ∫sec2x. cos2 2x dx

    = ∫ sec2x (2cos2x – 1)2 dx

    = ∫ (4cos2 x + sec2x- 4)dx

    =∫(2cos 2x + sec2x-2)dx

    = sin 2x + tan x – 2x

And I3 = ∫sin4x. cos4 x dx

          = 1/128 ∫(3 - 4cos 4x + cos 8x)dx

          = 3x/128 – sin 4x /128 + sin 8x/1024

Hence, I = I1 + I2 + I3

             = -cos 4x/16 -cos 2x/8 + cos 6x/24 + sin 2x + tan x – 2x + 3x/128 – sin 4x /128 + sin 8x/1024. 

Illustration 3: For any natural umber m, evaluate

∫(x3m + x2m + xm) (2x2m + 3xm + 6)1/mdx, x > 0.(2002)

Solution: For any natural number m, the given integral can be written as

I = ∫(x3m...

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