What is the integral of ((3Inx-5)^4)/x ?

The integral of ((3Inx-5)^4)/x is 625ln|x|-1500Inx+675I^2n^2x^2-180I^3n^3x^3+(81I^4n^4x^4)/4+C

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integral of ((3Inx-5)^4)/x

\int \frac{( 3 I n x - 5 ) ^{4}}{x} d x

625 ln ∣ x ∣ - 1500 I n x + 675 I^{2} n^{2} x^{2} - 180 I^{3} n^{3} x^{3} + \frac{81 I ^{4} n ^{4} x ^{4}}{4} + C

Solution steps

\int \frac{( 3 I n x - 5 ) ^{4}}{x} d x

Expand

\frac{( - 5 + 3 I n x ) ^{4}}{x} : \frac{625}{x} - 1500 I n + 1350 I^{2} n^{2} x - 540 I^{3} n^{3} x^{2} + 81 I^{4} n^{4} x^{3}

= \int \frac{625}{x} - 1500 I n + 1350 I^{2} n^{2} x - 540 I^{3} n^{3} x^{2} + 81 I^{4} n^{4} x^{3} 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 \frac{625}{x} d x - \int 1500 I n d x + \int 1350 I^{2} n^{2} x d x - \int 540 I^{3} n^{3} x^{2} d x + \int 81 I^{4} n^{4} x^{3} d x

\int \frac{625}{x} d x = 625 ln ∣ x ∣

\int 1500 I n d x = 1500 I n x

\int 1350 I^{2} n^{2} x d x = 675 I^{2} n^{2} x^{2}

\int 540 I^{3} n^{3} x^{2} d x = 180 I^{3} n^{3} x^{3}

\int 81 I^{4} n^{4} x^{3} d x = \frac{81 I ^{4} n ^{4} x ^{4}}{4}

= 625 ln ∣ x ∣ - 1500 I n x + 675 I^{2} n^{2} x^{2} - 180 I^{3} n^{3} x^{3} + \frac{81 I ^{4} n ^{4} x ^{4}}{4}

Add a constant to the solution

= 625 ln ∣ x ∣ - 1500 I n x + 675 I^{2} n^{2} x^{2} - 180 I^{3} n^{3} x^{3} + \frac{81 I ^{4} n ^{4} x ^{4}}{4} + C

\int \frac{7}{x ^{2} + 1} d x

\int_{6}^{4} (\frac{- 1}{2} x + 3) d x

x \to 0 lim (\frac{( ∣ x ∣ )}{x})

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

y^{^{''}} + 4 y = 3 csc^{2} (2 t)

What is the integral of ((3Inx-5)^4)/x ?
The integral of ((3Inx-5)^4)/x is 625ln|x|-1500Inx+675I^2n^2x^2-180I^3n^3x^3+(81I^4n^4x^4)/4+C