What is the integral of (t+1/t)^{3/2}((t^2-1)/(t^2)) ?

The integral of (t+1/t)^{3/2}((t^2-1)/(t^2)) is 2/5 ((t^2+1)/t)^{5/2}+C

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integral of (t+1/t)^{3/2}((t^2-1)/(t^2))

\int (t + \frac{1}{t})^{\frac{3}{2}} (\frac{t ^{2} - 1}{t ^{2}}) d t

\frac{2}{5} (\frac{t ^{2} + 1}{t})^{\frac{5}{2}} + C

Solution steps

\int (t + \frac{1}{t})^{\frac{3}{2}} \frac{t ^{2} - 1}{t ^{2}} d t

Apply u-substitution

: \int u^{\frac{3}{2}} d u

= \int u^{\frac{3}{2}} d u

Apply the Power Rule

: \frac{2}{5} u^{\frac{5}{2}}

= \frac{2}{5} u^{\frac{5}{2}}

Substitute back

u = t + \frac{1}{t}

= \frac{2}{5} (t + \frac{1}{t})^{\frac{5}{2}}

Join

t + \frac{1}{t} : \frac{t ^{2} + 1}{t}

= \frac{2}{5} (\frac{t ^{2} + 1}{t})^{\frac{5}{2}}

Add a constant to the solution

= \frac{2}{5} (\frac{t ^{2} + 1}{t})^{\frac{5}{2}} + C

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What is the integral of (t+1/t)^{3/2}((t^2-1)/(t^2)) ?
The integral of (t+1/t)^{3/2}((t^2-1)/(t^2)) is 2/5 ((t^2+1)/t)^{5/2}+C