What is the integral of ln^4(x)+1 ?

The integral of ln^4(x)+1 is xln^4(x)-4xln^3(x)+12xln^2(x)-24xln(x)+25x+C

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integral of ln^4(x)+1

integral

ln^{4} (x) + 1

x ln^{4} (x) - 4 x ln^{3} (x) + 12 x ln^{2} (x) - 24 x ln (x) + 25 x + C

Solution steps

\int ln^{4} (x) + 1 d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int ln^{4} (x) d x + \int 1 d x

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

\int 1 d x = x

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

Simplify

x ln^{4} (x) - 4 (x ln^{3} (x) - 3 (x ln^{2} (x) - 2 (x ln (x) - x))) + x : x ln^{4} (x) - 4 x ln^{3} (x) + 12 x ln^{2} (x) - 24 x ln (x) + 25 x

= x ln^{4} (x) - 4 x ln^{3} (x) + 12 x ln^{2} (x) - 24 x ln (x) + 25 x

Add a constant to the solution

= x ln^{4} (x) - 4 x ln^{3} (x) + 12 x ln^{2} (x) - 24 x ln (x) + 25 x + C

\frac{1}{z ^{2}}

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

3 - x^{2}

\frac{( x ^{2} + 1 )}{( x + 1 )}

(3^{x})^{2}

What is the integral of ln^4(x)+1 ?
The integral of ln^4(x)+1 is xln^4(x)-4xln^3(x)+12xln^2(x)-24xln(x)+25x+C