حل ∫ (from 2 to 100) of x^3+3y

$$\int _ { 2 } ^ { 100 } x ^ { 3 } + 3 y$$

تقدير القيمة

$$294\left(y+85034\right)$$

خطوات الحل

Evaluate the indefinite integral first.

$$\int x^{3}+3y\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{3}\mathrm{d}x+\int 3y\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int x^{3}\mathrm{d}x+3\int y\mathrm{d}x$$

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}.

$$\frac{x^{4}}{4}+3\int y\mathrm{d}x$$

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

$$\frac{x^{4}}{4}+3yx$$

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.

$$\frac{100^{4}}{4}+3y\times 100-\left(\frac{2^{4}}{4}+3y\times 2\right)$$

تبسيط.

$$24999996+294y$$

تفاضل w.r.t. y

$$294$$