What is the integral of t^2sqrt(1+1/(t^2)) ?

The integral of t^2sqrt(1+1/(t^2)) is 1/3 t^2sqrt(t^2+1)+1/3 sqrt(t^2+1)+C

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integral of t^2sqrt(1+1/(t^2))

\int t^{2} 1 + \frac{1}{t ^{2}} d t

\frac{1}{3} t^{2} t^{2} + 1 + \frac{1}{3} t^{2} + 1 + C

Solution steps

\int t^{2} 1 + \frac{1}{t ^{2}} d t

Apply Integration By Parts

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

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

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

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

Simplify

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

= \frac{1}{3} t^{2} t^{2} + 1 + \frac{1}{3} t^{2} + 1

Add a constant to the solution

= \frac{1}{3} t^{2} t^{2} + 1 + \frac{1}{3} t^{2} + 1 + C

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What is the integral of t^2sqrt(1+1/(t^2)) ?
The integral of t^2sqrt(1+1/(t^2)) is 1/3 t^2sqrt(t^2+1)+1/3 sqrt(t^2+1)+C