What is the integral from 1 to x^3 of 1/(1+ln(t)) ?

The integral from 1 to x^3 of 1/(1+ln(t)) is 1/e (\Ei(1+3ln(x))-\Ei(1))

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integral from 1 to x^3 of 1/(1+ln(t))

\int_{1}^{x^{3}} \frac{1}{1 + ln ( t )} d t

\frac{1}{e} (Ei (1 + 3 ln (x)) - Ei (1))

Solution steps

\int_{1}^{x^{3}} \frac{1}{1 + ln ( t )} d t

Apply u-substitution

: \int_{1}^{1 + l n (x^{3})} \frac{e ^{u - 1}}{u} d u

= \int_{1}^{1 + l n (x^{3})} \frac{e ^{u - 1}}{u} d u

Simplify

e^{u - 1} : e^{u} e^{- 1}

= \int_{1}^{1 + l n (x^{3})} \frac{e ^{u} e ^{- 1}}{u} d u

Take the constant out:

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

= e^{- 1} \cdot \int_{1}^{1 + l n (x^{3})} \frac{e ^{u}}{u} d u

Apply exponent rule:

a^{- 1} = \frac{1}{a}

= \frac{1}{e} \cdot \int_{1}^{1 + l n (x^{3})} \frac{e ^{u}}{u} d u

Use the nonelementary integral:

\int \frac{e ^{x}}{x} d x = Ei (x)

= \frac{1}{e} [Ei (u)]_{1}^{1 + l n (x^{3})}

Apply log rule

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x),

assuming

x \geq 0

= \frac{1}{e} [Ei (u)]_{1}^{1 + 3 l n (x)}

Compute the boundaries:

Ei (1 + 3 ln (x)) - Ei (1)

= \frac{1}{e} (Ei (1 + 3 ln (x)) - Ei (1))

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What is the integral from 1 to x^3 of 1/(1+ln(t)) ?
The integral from 1 to x^3 of 1/(1+ln(t)) is 1/e (\Ei(1+3ln(x))-\Ei(1))