Solve ∫ (from 2 to 233) of yx^2z

$$\int _ { 2 } ^ { 233 } y x ^ { 2 } z$$

Evaluate

$$4216443yz$$

Solution Steps

Evaluate the indefinite integral first.

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

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

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

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}.

$$yz\times \frac{x^{3}}{3}$$

Simplify.

$$\frac{yzx^{3}}{3}$$

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.

$$\frac{1}{3}yz\times 233^{3}-\frac{1}{3}yz\times 2^{3}$$

Simplify.

$$4216443yz$$

Differentiate w.r.t. y

$$4216443z$$