Solve ∫ (from 1 to 4) of 1/sqrt{t(sqrt{t}+1)^3}

$$\int _ { 1 } ^ { 4 } \frac { 1 } { \sqrt { t } ( \sqrt { t } + 1 ) ^ { 3 } }$$

Evaluate

$$\frac{3}{\sqrt{t}\left(\sqrt{t}+1\right)^{3}}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{1}{\sqrt{t}\left(\sqrt{t}+1\right)^{3}}\mathrm{d}x$$

Find the integral of \frac{1}{\sqrt{t}\left(\sqrt{t}+1\right)^{3}} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{x}{\sqrt{t}\left(\sqrt{t}+1\right)^{3}}$$

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.

$$t^{-\frac{1}{2}}\left(t^{\frac{1}{2}}+1\right)^{-3}\times 4-t^{-\frac{1}{2}}\left(t^{\frac{1}{2}}+1\right)^{-3}$$

Simplify.

$$\frac{3}{\sqrt{t}\left(\sqrt{t}+1\right)^{3}}$$

Differentiate w.r.t. t

$$\frac{3\left(-4\sqrt{t}-1\right)}{2t^{\frac{3}{2}}\left(\sqrt{t}+1\right)^{4}}$$