What is the integral of (2pi(ln(t))^2)/(sqrt(t)) ?

The integral of (2pi(ln(t))^2)/(sqrt(t)) is 2pi(2sqrt(t)ln^2(t)-4(2sqrt(t)ln(t)-4sqrt(t)))+C

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integral of (2pi(ln(t))^2)/(sqrt(t))

\int \frac{2 π ( ln ( t ) ) ^{2}}{t} d t

2 π (2 t ln^{2} (t) - 4 (2 t ln (t) - 4 t)) + C

Solution steps

\int \frac{2 π ( ln ( t ) ) ^{2}}{t} d t

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 π \cdot \int \frac{( ln ( t ) ) ^{2}}{t} d t

Simplify

= 2 π \cdot \int \frac{ln ^{2} ( t )}{t} d t

Apply Integration By Parts

: 2 t ln^{2} (t) - \int \frac{4 ln ( t )}{t} d t

= 2 π (2 t ln^{2} (t) - \int \frac{4 ln ( t )}{t} d t)

\int \frac{4 ln ( t )}{t} d t = 4 (2 t ln (t) - 4 t)

= 2 π (2 t ln^{2} (t) - 4 (2 t ln (t) - 4 t))

Add a constant to the solution

= 2 π (2 t ln^{2} (t) - 4 (2 t ln (t) - 4 t)) + C

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What is the integral of (2pi(ln(t))^2)/(sqrt(t)) ?
The integral of (2pi(ln(t))^2)/(sqrt(t)) is 2pi(2sqrt(t)ln^2(t)-4(2sqrt(t)ln(t)-4sqrt(t)))+C