Solve ∫ (from - a to a) of x[f(x)+f(-x)] wrt x

$$\int _ { - a } ^ { a } x [ f ( x ) + f ( - x ) ] d x$$

Evaluate

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Solution Steps

Use the distributive property to multiply x by fx+f\left(-x\right).

$$\int _{-a}^{a}fx^{2}+xf\left(-x\right)\mathrm{d}x$$

Multiply x and x to get x^{2}.

$$\int _{-a}^{a}fx^{2}+x^{2}f\left(-1\right)\mathrm{d}x$$

Combine fx^{2} and x^{2}f\left(-1\right) to get 0.

$$\int _{-a}^{a}0\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 0\mathrm{d}x$$

Find the integral of 0 using the table of common integrals rule \int a\mathrm{d}x=ax.

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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.

$$0+0$$

Simplify.

$$0$$

Differentiate w.r.t. f

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