What is the integral of ([ln(x)+7]^3)/x ?

The integral of ([ln(x)+7]^3)/x is ((ln(x)+7)^4)/4+C

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integral of ([ln(x)+7]^3)/x

\int \frac{[ ln ( x ) + 7 ] ^{3}}{x} d x

\frac{( ln ( x ) + 7 ) ^{4}}{4} + C

Solution steps

\int \frac{[ ln ( x ) + 7 ] ^{3}}{x} d x

Simplify

(ln (x) + 7)^{3} : (ln (x) + 7)^{2} (ln (x) + 7)

= \int \frac{( ln ( x ) + 7 ) ^{2} ( ln ( x ) + 7 )}{x} d x

(ln (x) + 7)^{2} (ln (x) + 7) = (ln (x) + 7)^{3}

= \int \frac{( ln ( x ) + 7 ) ^{3}}{x} d x

Apply u-substitution

: \int u^{3} d u

= \int u^{3} d u

Apply the Power Rule

: \frac{u ^{4}}{4}

= \frac{u ^{4}}{4}

Substitute back

u = ln (x) + 7

= \frac{( ln ( x ) + 7 ) ^{4}}{4}

Add a constant to the solution

= \frac{( ln ( x ) + 7 ) ^{4}}{4} + C

\int \frac{r ln ( r ^{2} + 1 )}{r ^{2} + 1} d r

\int \frac{4 x}{( x + 2 ) ^{2}} d x

\int y^{2} 1 - y^{2} d y

\int (x + 1) (x^{2} + 2 x)^{3} d x

What is the integral of ([ln(x)+7]^3)/x ?
The integral of ([ln(x)+7]^3)/x is ((ln(x)+7)^4)/4+C