Solve ∫ (from 1 to 9) of 4x/x^1,5

$$\int _ { 1 } ^ { 9 } \frac { 4 x } { x ^ { 1,5 } }$$

Evaluate

$$16$$

Solution Steps

Cancel out x in both numerator and denominator.

$$\int _{1}^{9}\frac{4}{\sqrt{x}}\mathrm{d}x$$

Evaluate the indefinite integral first.

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

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

$$4\int \frac{1}{\sqrt{x}}\mathrm{d}x$$

Rewrite \frac{1}{\sqrt{x}} as x^{-\frac{1}{2}}. Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{-\frac{1}{2}}\mathrm{d}x with \frac{x^{\frac{1}{2}}}{\frac{1}{2}}. Simplify and convert from exponential to radical form.

$$8\sqrt{x}$$

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.

$$8\times 9^{0.5}-8\times 1^{0.5}$$

Simplify.

$$16$$