Solve ∫ (from 0 to 2) of (4x^2+5y) wrt x

$$\int _ { 0 } ^ { 2 } ( 4 x ^ { 2 } + 5 y ) d x$$

Evaluate

$$10y+\frac{32}{3}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 4x^{2}+5y\mathrm{d}x$$

Integrate the sum term by term.

$$\int 4x^{2}\mathrm{d}x+\int 5y\mathrm{d}x$$

Factor out the constant in each of the terms.

$$4\int x^{2}\mathrm{d}x+5\int y\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}. Multiply 4 times \frac{x^{3}}{3}.

$$\frac{4x^{3}}{3}+5\int y\mathrm{d}x$$

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

$$\frac{4x^{3}}{3}+5yx$$

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.

$$\frac{4}{3}\times 2^{3}+5y\times 2-\left(\frac{4}{3}\times 0^{3}+5y\times 0\right)$$

Simplify.

$$\frac{32}{3}+10y$$

Differentiate w.r.t. y

$$10$$