What is the integral of ln^5(x^2) ?

The integral of ln^5(x^2) is 32(xln^5(x)-5(xln^4(x)-4(xln^3(x)-3(xln^2(x)-2(xln(x)-x)))))+C

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integral of ln^5(x^2)

integral

ln^{5} (x^{2})

32 (x ln^{5} (x) - 5 (x ln^{4} (x) - 4 (x ln^{3} (x) - 3 (x ln^{2} (x) - 2 (x ln (x) - x))))) + C

Solution steps

\int ln^{5} (x^{2}) d x

Simplify

ln^{5} (x^{2}) : 32 ln^{5} (x)

= \int 32 ln^{5} (x) d x

Take the constant out:

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

= 32 \cdot \int ln^{5} (x) d x

Apply Integration By Parts

: x ln^{5} (x) - \int 5 ln^{4} (x) d x

= 32 (x ln^{5} (x) - \int 5 ln^{4} (x) d x)

\int 5 ln^{4} (x) d x = 5 (x ln^{4} (x) - 4 (x ln^{3} (x) - 3 (x ln^{2} (x) - 2 (x ln (x) - x))))

= 32 (x ln^{5} (x) - 5 (x ln^{4} (x) - 4 (x ln^{3} (x) - 3 (x ln^{2} (x) - 2 (x ln (x) - x)))))

Add a constant to the solution

= 32 (x ln^{5} (x) - 5 (x ln^{4} (x) - 4 (x ln^{3} (x) - 3 (x ln^{2} (x) - 2 (x ln (x) - x))))) + C

\frac{x ^{3}}{x ^{2}}

(e^{x})^{2} x

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

1 + e^{v}

y^{2} ((1 - y^{2})^{\frac{1}{2}})

What is the integral of ln^5(x^2) ?
The integral of ln^5(x^2) is 32(xln^5(x)-5(xln^4(x)-4(xln^3(x)-3(xln^2(x)-2(xln(x)-x)))))+C