Solve ∫ (from 2 to 3) of 4x^3/x^2 wrt x

$$\int _ { 2 } ^ { 3 } \frac { 4 x ^ { 3 } } { x ^ { 2 } } d x$$

Evaluate

$$10$$

Solution Steps

Cancel out x^{2} in both numerator and denominator.

$$\int _{2}^{3}4x\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 4x\mathrm{d}x$$

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

$$4\int x\mathrm{d}x$$

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 4 times \frac{x^{2}}{2}.

$$2x^{2}$$

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.

$$2\times 3^{2}-2\times 2^{2}$$

Simplify.

$$10$$