حل ∫ 5x(4-2x^2)

$$\int 5 x ( 4 - 2 x ^ { 2 } )$$

تقدير القيمة

$$-\frac{5x^{4}}{2}+10x^{2}+С$$

خطوات الحل

استخدم خاصية التوزيع لضرب 5x في 4-2x^{2}.

$$\int 20x-10x^{3}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 20x\mathrm{d}x+\int -10x^{3}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$20\int x\mathrm{d}x-10\int x^{3}\mathrm{d}x$$

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}. اضرب 20 في \frac{x^{2}}{2}.

$$10x^{2}-10\int x^{3}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}. اضرب -10 في \frac{x^{4}}{4}.

$$10x^{2}-\frac{5x^{4}}{2}$$

If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$10x^{2}-\frac{5x^{4}}{2}+С$$

تفاضل w.r.t. x

$$20x-10x^{3}$$