Solve ∫ (from 1 to 2) of (x^2z) wrt z

$$\int_{ 1 }^{ 2 } ( { x }^{ 2 } z) d z$$

Evaluate

$$\frac{3x^{2}}{2}$$

Solution Steps

Evaluate the indefinite integral first.

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

Factor out the constant using \int af\left(z\right)\mathrm{d}z=a\int f\left(z\right)\mathrm{d}z.

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

Since \int z^{k}\mathrm{d}z=\frac{z^{k+1}}{k+1} for k\neq -1, replace \int z\mathrm{d}z with \frac{z^{2}}{2}.

$$x^{2}\times \frac{z^{2}}{2}$$

Simplify.

$$\frac{x^{2}z^{2}}{2}$$

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.

$$\frac{1}{2}x^{2}\times 2^{2}-\frac{1}{2}x^{2}\times 1^{2}$$

Simplify.

$$\frac{3x^{2}}{2}$$

Differentiate w.r.t. x

$$3x$$