2. Evaluate the following integrals (a) ∫0^∞(x sin x)/((x^2...

2. Evaluate the following integrals (a) \( \int_0^\infty \frac{x \sin x}{(x^2 + 1)^2} dx \) (25 points) (b) \( \int_0^\infty \frac{x^2}{(x^2 + a^2)^2} dx, a > 0 \) (25 points) (c) \( \int_0^\infty \frac{x^a}{x^2 + 1} dx, -1 < a < 1 \) (use branch cut: 30

Transcript

00:01 Hi, today we are solving the question in which we need to evaluate firstly integral from 1 to 2 of 6x square minus 1 whole into dx.

00:11 Now here applying the sum rule and solving it.

00:15 So integral from 1 to 2 of 6x square into dx minus integral from 1 to 2 of 1 into dx.

00:26 Now solving it we get 6x cube by 3 and limits are from 1 to 2 minus x and limits are from 1 to 2.

00:45 Now 3, 1 and here it is 2.

00:48 So 2x cube will become here after putting the limits 14 minus 1.

00:55 So we get it as equals to 13.

00:57 So we can conclude that therefore required answer is equals to 13.

01:05 Now taking the second one we need to evaluate the integral from 0 to 1 of x raised to the power 5 minus 1 into x raised to the power 5 plus 1 whole into dx.

01:19 Now from here multiplying the conjugate.

01:24 So integral from 0 to 1 of x raised to the power 10 minus 1 divided by x raised to the power 5 plus 1 whole into x plus x raised to the power 5 plus 1 into dx.

01:37 Now simplifying and solving it.

01:40 So we get integral from 0 to 1 x raised to the power 10 minus 1 into dx.

01:47 On solving it we get integral from x raised to the power 10 minus 1 will become 11.

02:04 From here x raised to the power 11 by 11 plus not plus it is minus x and limits are from 0 to 1.

02:22 So from here on solving it we get it as minus 10 by 11.

02:29 So therefore required answer is minus 10 by 11.

02:38 Now similarly taking the next one.

02:41 So here we are given with integral from 1 to 2 of x square into under root x cube plus 8 whole into dx...

Question

2. Evaluate the following integrals (a) \( \int_0^\infty \frac{x \sin x}{(x^2 + 1)^2} dx \) (25 points) (b) \( \int_0^\infty \frac{x^2}{(x^2 + a^2)^2} dx, a > 0 \) (25 points) (c) \( \int_0^\infty \frac{x^a}{x^2 + 1} dx, -1 < a < 1 \) (use branch cut: 30

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