What is the integral from 0 to pi of ((sin(x))/x)^2 ?

The integral from 0 to pi of ((sin(x))/x)^2 is diverges

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integral from 0 to pi of ((sin(x))/x)^2

\int_{0}^{π} (\frac{sin ( x )}{x})^{2} d x

diverges

Solution steps

\int_{0}^{π} (\frac{sin ( x )}{x})^{2} d x

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \int_{0}^{π} \frac{sin ^{2} ( x )}{x ^{2}} d x

Rewrite using trig identities

= \int_{0}^{π} \frac{1 - cos ^{2} ( x )}{x ^{2}} d x

Expand

\frac{1 - cos ^{2} ( x )}{x ^{2}} : \frac{1}{x ^{2}} - \frac{cos ^{2} ( x )}{x ^{2}}

= \int_{0}^{π} \frac{1}{x ^{2}} - \frac{cos ^{2} ( x )}{x ^{2}} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{0}^{π} \frac{1}{x ^{2}} d x - \int_{0}^{π} \frac{cos ^{2} ( x )}{x ^{2}} d x

\int_{0}^{π} \frac{1}{x ^{2}} d x =

diverges

If part of the expression diverges, then entire expression diverges

= diverges

\int_{0}^{3} \frac{x ^{2}}{2} d x

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\int_{0}^{1} 20 (x^{3} - x^{4}) d x

\int_{0}^{1} (5 x^{4} - 3 x^{2} + 1) d x

What is the integral from 0 to pi of ((sin(x))/x)^2 ?
The integral from 0 to pi of ((sin(x))/x)^2 is diverges