Solve ∫ (from 0 to sqrt { y) of }(x^2-3)

$$\int _ { 0 } ^ { \sqrt { y } } ( x ^ { 2 } - 3 )$$

Evaluate

$$\frac{\sqrt{y}\left(y-9\right)}{3}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x^{2}-3\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{2}\mathrm{d}x+\int -3\mathrm{d}x$$

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}.

$$\frac{x^{3}}{3}+\int -3\mathrm{d}x$$

Find the integral of -3 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{x^{3}}{3}-3x$$

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.

$$\frac{1}{3}\left(y^{\frac{1}{2}}\right)^{3}-3y^{\frac{1}{2}}-\left(\frac{0^{3}}{3}-3\times 0\right)$$

Simplify.

$$\frac{\left(-9+y\right)\sqrt{y}}{3}$$

Differentiate w.r.t. y

$$\frac{y-3}{2\sqrt{y}}$$