Solve ∫ (from 0 to 1) of y+5x) wrt x

$$\int _ { 0 } ^ { 1 } y + 5 x ) d x$$

Evaluate

$$y = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

Solution Steps

Evaluate the indefinite integral first.

$$\int y+5x\mathrm{d}x$$

Integrate the sum term by term.

$$\int y\mathrm{d}x+\int 5x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int y\mathrm{d}x+5\int x\mathrm{d}x$$

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

$$yx+5\int x\mathrm{d}x$$

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}. Multiply 5 times \frac{x^{2}}{2}.

$$yx+\frac{5x^{2}}{2}$$

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.

$$y\times 1+\frac{5}{2}\times 1^{2}-\left(y\times 0+\frac{5}{2}\times 0^{2}\right)$$

Simplify.

$$y+\frac{5}{2}$$

Differentiate w.r.t. y

$$1$$