Solve ∫ _{-sqrt{pi}}^sqrt{pi}f(x) wrt x

$$\int _ { - \sqrt { \pi } } ^ { \sqrt { \pi } } f ( x ) d x$$

Evaluate

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Solution Steps

Evaluate the indefinite integral first.

$$\int fx\mathrm{d}x$$

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$f\int x\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}.

$$f\times \frac{x^{2}}{2}$$

Simplify.

$$\frac{fx^{2}}{2}$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$\frac{1}{2}f\left(\pi ^{\frac{1}{2}}\right)^{2}-\frac{1}{2}f\left(-\pi ^{\frac{1}{2}}\right)^{2}$$

Simplify.

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Differentiate w.r.t. f

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