What is the integral of 1/(xsqrt(1+ln(x^5))) ?

The integral of 1/(xsqrt(1+ln(x^5))) is 2/5 sqrt(1+5ln(x))+C

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integral of 1/(xsqrt(1+ln(x^5)))

\int \frac{1}{x 1 + ln ( x ^{5} )} d x

\frac{2}{5} 1 + 5 ln (x) + C

Solution steps

\int \frac{1}{x 1 + ln ( x ^{5} )} d x

1 + ln (x^{5}) = 1 + 5 ln (x)

= \int \frac{1}{x 1 + 5 ln ( x )} d x

Apply u-substitution

= \int \frac{1}{5 u} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int \frac{1}{u} d u

Apply the Power Rule

= \frac{1}{5} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 1 + 5 ln (x)

= \frac{1}{5} \cdot 2 (1 + 5 ln (x))^{\frac{1}{2}}

Simplify

\frac{1}{5} \cdot 2 (1 + 5 ln (x))^{\frac{1}{2}} : \frac{2}{5} 1 + 5 ln (x)

= \frac{2}{5} 1 + 5 ln (x)

Add a constant to the solution

= \frac{2}{5} 1 + 5 ln (x) + C

What is the integral of 1/(xsqrt(1+ln(x^5))) ?
The integral of 1/(xsqrt(1+ln(x^5))) is 2/5 sqrt(1+5ln(x))+C