Solve ∫ (from 1 to 5) of (z-x^2) wrt x

$$\int _ { 1 } ^ { 5 } ( z - x ^ { 2 } ) d x$$

Evaluate

$$4z-\frac{124}{3}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int z-x^{2}\mathrm{d}x$$

Integrate the sum term by term.

$$\int z\mathrm{d}x+\int -x^{2}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int z\mathrm{d}x-\int x^{2}\mathrm{d}x$$

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

$$zx-\int x^{2}\mathrm{d}x$$

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 -1 times \frac{x^{3}}{3}.

$$zx-\frac{x^{3}}{3}$$

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.

$$z\times 5-\frac{5^{3}}{3}-\left(z\times 1-\frac{1^{3}}{3}\right)$$

Simplify.

$$-\frac{124}{3}+4z$$

Differentiate w.r.t. z

$$4$$