Solve ∫ (from y to 2) of 4/5x^2 wrt x

$$\int _ { y } ^ { 2 } 4 / 5 x ^ { 2 } d x$$

Evaluate

$$\frac{32-4y^{3}}{15}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{4x^{2}}{5}\mathrm{d}x$$

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

$$\frac{4\int x^{2}\mathrm{d}x}{5}$$

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}.

$$\frac{4x^{3}}{15}$$

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.

$$\frac{4}{15}\times 2^{3}-\frac{4}{15}y^{3}$$

Simplify.

$$\frac{-4y^{3}+32}{15}$$

Differentiate w.r.t. y

$$-\frac{4y^{2}}{5}$$