Solve ∫ (from 1 to 4) of sqrt{1-frac{24sqrt{y}-1}{4y}+36}

$$\int _ { 1 } ^ { 4 } \sqrt { 1 - \frac { 24 \sqrt { y } - 1 } { 4 y } + 36 }$$

Evaluate

$$\frac{3\sqrt{148-\frac{24}{\sqrt{y}}+\frac{1}{y}}}{2}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \sqrt{1-\frac{24\sqrt{y}-1}{4y}+36}\mathrm{d}x$$

Find the integral of \sqrt{1-\frac{24\sqrt{y}-1}{4y}+36} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\sqrt{1-\frac{24\sqrt{y}-1}{4y}+36}x$$

Simplify.

$$\frac{\sqrt{148-\frac{24}{\sqrt{y}}+\frac{1}{y}}x}{2}$$

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.

$$\frac{1}{2}\left(148-24y^{-\frac{1}{2}}+y^{-1}\right)^{\frac{1}{2}}\times 4-\frac{1}{2}\left(148-24y^{-\frac{1}{2}}+y^{-1}\right)^{\frac{1}{2}}$$

Simplify.

$$\frac{3\sqrt{148y-24\sqrt{y}+1}}{2\sqrt{y}}$$

Differentiate w.r.t. y

$$\frac{3\left(-\sqrt{y\left(148y-24\sqrt{y}+1\right)^{2}}+4\left(37\sqrt{y}-3\right)y\right)}{4y^{\frac{3}{2}}\sqrt{y\left(148y-24\sqrt{y}+1\right)}}$$