Solve ∫ (from 5 / 4 to 2) of 1/x^2y4/15x^2 wrt x

$$\int _ { 5 / 4 } ^ { 2 } \frac { 1 } { x ^ { 2 } y } \frac { 4 } { 15 } x ^ { 2 } d x$$

Evaluate

$$\frac{1}{5y}$$

Solution Steps

Multiply \frac{1}{x^{2}y} times \frac{4}{15} by multiplying numerator times numerator and denominator times denominator.

$$\int _{\frac{5}{4}}^{2}\frac{4}{x^{2}y\times 15}x^{2}\mathrm{d}x$$

Express \frac{4}{x^{2}y\times 15}x^{2} as a single fraction.

$$\int _{\frac{5}{4}}^{2}\frac{4x^{2}}{x^{2}y\times 15}\mathrm{d}x$$

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

$$\int _{\frac{5}{4}}^{2}\frac{4}{15y}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int \frac{4}{15y}\mathrm{d}x$$

Find the integral of \frac{4}{15y} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{4}{15y}x$$

Simplify.

$$\frac{4x}{15y}$$

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.

$$\frac{4}{15}y^{-1}\times 2-\frac{4}{15}y^{-1}\times \frac{5}{4}$$

Simplify.

$$\frac{1}{5y}$$

Differentiate w.r.t. y

$$-\frac{1}{5y^{2}}$$