Integral of the year and my 50 followers problem

Calculus Level 4

∫ x 4 d x ( x 2 + 100 7 2 ) ( x 2 + 100 8 2 ) = ? \large \displaystyle \int \dfrac{x^4 dx}{(x^2 + 1007^2)(x^2 + 1008^2)} = \ ?

x + 100 7 3 2015 tan ⁡ − 1 x 1007 + 100 8 3 2015 tan ⁡ − 1 x 1008 + C x + \frac{1007^3}{2015}\tan^{-1}\frac{x}{1007} + \frac{1008^3}{2015}\tan^{-1}\frac{x}{1008} + C None of these x + 1007 2015 tan ⁡ − 1 x 1007 + 1008 2015 tan ⁡ − 1 x 1008 + C x + \frac{1007}{2015}\tan^{-1}\frac{x}{1007} + \frac{1008}{2015}\tan^{-1}\frac{x}{1008} + C x + 100 7 3 2015 tan ⁡ − 1 x 1007 − 100 8 3 2015 tan ⁡ − 1 x 1008 + C x + \frac{1007^3}{2015}\tan^{-1}\frac{x}{1007} - \frac{1008^3}{2015}\tan^{-1}\frac{x}{1008} + C x + 100 7 2 2015 tan ⁡ − 1 x 1007 − 100 8 2 2015 tan ⁡ − 1 x 1008 + C x + \frac{1007^2}{2015}\tan^{-1}\frac{x}{1007} - \frac{1008^2}{2015}\tan^{-1}\frac{x}{1008} + C

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Vishwak Srinivasan by Vishwak Srinivasan , 24, India

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1 solution

Let the given integral be denoted by I I .

I = ∫ ( ( x 2 + 100 7 2 ) − 100 7 2 ) ( ( x 2 + 100 8 2 ) − 100 8 2 ) d x ( x 2 + 100 7 2 ) ( x 2 + 100 8 2 ) I = \displaystyle \int \frac{((x^2 + 1007^2) - 1007^2)((x^2 + 1008^2) - 1008^2)dx}{(x^2 + 1007^2)(x^2 + 1008^2)}

I = ∫ d x − 100 7 2 ∫ d x x 2 + 100 7 2 − 100 8 2 ∫ d x x 2 + 100 8 2 + ( 1007 × 1008 ) 2 ∫ d x ( x 2 + 100 7 2 ) ( x 2 + 100 8 2 ) I = \displaystyle \int dx - 1007^2 \int \frac{dx}{x^2 + 1007^2} - 1008^2 \int \frac{dx}{x^2 + 1008^2} + (1007 \times 1008)^2 \int \frac{dx}{(x^2 + 1007^2)(x^2 +1008^2)}

I = x − 100 7 2 1 1007 tan ⁡ − 1 ( x 1007 ) − 100 8 2 1 1008 tan ⁡ − 1 ( x 1008 ) + ( 1007 × 1008 ) 2 100 8 2 − 100 7 2 ∫ ( 1 x 2 + 100 7 2 − 1 x 2 + 100 8 2 ) d x I = \displaystyle x - 1007^2 \frac{1}{1007}\tan^{-1}(\frac{x}{1007}) - 1008^2\frac{1}{1008}\tan^{-1}(\frac{x}{1008}) + \frac{(1007\times 1008)^2}{1008^2 - 1007^2} \int (\frac{1}{x^2 + 1007^2} - \frac{1}{x^2 + 1008^2}) dx

I = x − 1007 tan ⁡ − 1 ( x 1007 ) − 1008 tan ⁡ − 1 ( x 1008 ) + ( 1007 × 1008 ) 2 2015 × ( 1 1007 tan ⁡ − 1 ( x 1007 ) − 1 1008 tan ⁡ − 1 ( x 1008 ) ) + C I = x - 1007\tan^{-1}(\frac{x}{1007}) - 1008\tan^{-1}(\frac{x}{1008}) + \frac{(1007\times1008)^2}{2015} \times (\frac{1}{1007}\tan^{-1}(\frac{x}{1007}) - \frac{1}{1008}\tan^{-1}(\frac{x}{1008})) + C

Simplification is a little tedious, after which we get,

I = x + 100 7 3 2015 tan ⁡ − 1 ( x 1007 ) − 100 8 3 2015 tan ⁡ − 1 ( x 1008 ) + C I = \boxed{x + \frac{1007^3}{2015}\tan^{-1}(\frac{x}{1007}) - \frac{1008^3}{2015}\tan^{-1}(\frac{x}{1008}) + C}

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Moderator note:

Great! What would be indefinite integral be if x 4 x^4 is replaced with x 5 x^5 instead?

take x common and proceed in a similar way after which introduce x in each expression and solve accordingly

Priyesh Pandey - 6 years, 1 month ago

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Yep. You got that right.

Vishwak Srinivasan - 6 years, 1 month ago

What do you mean? I did the "let y = x 2 y=x^2 " thing.

Pi Han Goh - 6 years ago

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What i meant is to proceed in a similar fashion as with x^4, and when u reach at step 2 (see Vishwak Srinivasan's solution) then multiply whole thing with x. each of the integral is of known form here, so u can proceed in a regular fashion. I'll try to post the complete solution soon for x^5.

Priyesh Pandey - 6 years ago

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