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**Illustration 1: Integrate the curve x/ (1+x ^{4}). (1978)**

= 1/2.âˆ« 2x / (1 + (x

Put x

Hence, I = âˆ« du/ 2(1 + u

= 1/2 tan

= 1/2 tan

**Illustration 2: Integrate sin x. sin 2x.sin 3x + sec ^{2}x. cos^{2} 2x + sin^{4} x cos^{4} x. (1979)**

= 1/4 âˆ« sin 4x + sin 2x - sin 6x)dx

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

I

= âˆ« sec

= âˆ« (4cos

=âˆ«(2cos 2x + sec

= sin 2x + tan x â€“ 2x

And I

= 1/128 âˆ«(3 - 4cos 4x + cos 8x)dx

= 3x/128 â€“ sin 4x /128 + sin 8x/1024

Hence, I = I

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

**Illustration 3: For any natural umber m, evaluate****âˆ«(x ^{3m} + x^{2m }+ x^{m}) (2x^{2}^{m} + 3x^{m} + 6)^{1/m}dx, x > 0.(2002)**

I = âˆ«(x

Hence I = âˆ«(2x

Put (2x

Then 6x

Hence I = âˆ«t

= [(2x

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