Chapter 4: Q14P (page 237)

Given that $\int_{0}^{\infty} - \frac{d x}{y^{2} + x^{2}} = \frac{π}{2 y}$ , differentiate with respect to y and so evaluate $\int_{0}^{\infty} - \frac{dx}{y^{2} + x^{2}} .$

Short Answer

Expert verified

The value of $\int_{0}^{\infty} - \frac{dx}{y^{2} + x^{2}}$ is $\frac{π}{4 y^{3}}$ .

Step by step solution

Given Information

Given that $\int_{0}^{\infty} - \frac{dx}{y^{2} + x^{2}} = \frac{π}{2 y}$ .

Formula Used

We know that $\frac{d}{d x} \int_{u (x)}^{v (x)} f (x, t) d t = f (x, v) \frac{d v}{d x} - f (x, u) \frac{d u}{d x} + \int_{u}^{v} \frac{\partial f}{\partial x} d t$ .

Evaluate ∫0∞-dx(y2+x2).

$\int_{0}^{\infty} \frac{d x}{y^{2} + x^{2}} d x = \frac{1}{y^{2} + \infty^{2}} . \frac{d}{d y} (\infty) - \frac{1}{y^{2} + 0^{2}} . \frac{d}{d y} (0) + \int \frac{- 1}{(y^{2} + x^{2})} (2 y) d x = - 2 y \int_{0}^{\infty} \frac{d x}{(y^{2} + x^{2})}$

But,

$\int_{0}^{\infty} \frac{d x}{y^{2} + x^{2}} d x = \frac{π}{2 y} \int_{0}^{\infty} \frac{d x}{y^{2} + x^{2}} d x = \frac{d}{d y} (\frac{π}{2 y}) = - \frac{π}{2 y^{2}}$

Therefore,

$\frac{- π}{2 y^{2}} - = - 2 y \int_{0}^{\infty} \frac{d x}{{(y^{2} + x^{2})}^{2}} \int_{0}^{\infty} \frac{d x}{{(y^{2} + x^{2})}^{2}} = \frac{- π}{(2 y^{2}) (- 2 y^{2})} \int_{0}^{\infty} \frac{d x}{{(y^{2} + x^{2})}^{2}} = \frac{π}{4 y^{3}}$

Hence $\int_{0}^{\infty} \frac{d x}{{(y^{2} + x^{2})}^{2}} = \frac{π}{4 y^{3}}$ .

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Given that $\int_{0}^{\infty} - \frac{d x}{y^{2} + x^{2}} = \frac{π}{2 y}$ , differentiate with respect to y and so evaluate $\int_{0}^{\infty} - \frac{dx}{y^{2} + x^{2}} .$

Short Answer

Step by step solution

Given Information

Formula Used

Evaluate ∫0∞-dx(y2+x2).

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Given that ∫0∞-dxy2+x2=π2y, differentiate with respect to y and so evaluate∫0∞-dxy2+x2.

Short Answer

Step by step solution

Given Information

Formula Used

Evaluate ∫0∞-dx(y2+x2).

One App. One Place for Learning.

Most popular questions from this chapter

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Given that $\int_{0}^{\infty} - \frac{d x}{y^{2} + x^{2}} = \frac{π}{2 y}$ , differentiate with respect to y and so evaluate $\int_{0}^{\infty} - \frac{dx}{y^{2} + x^{2}} .$