What is the integral from e to e^8 of 1/(xln(x^7)) ?

The integral from e to e^8 of 1/(xln(x^7)) is 3/7 ln(2)

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integral from e to e^8 of 1/(xln(x^7))

\int_{e}^{e^{8}} \frac{1}{x ln ( x ^{7} )} d x

\frac{3}{7} ln (2)

Decimal Notation

0.29706 \dots

Solution steps

\int_{e}^{e^{8}} \frac{1}{x ln ( x ^{7} )} d x

Apply u-substitution

: \int_{e^{7}}^{e^{56}} \frac{1}{7 u ln ( u )} d u

= \int_{e^{7}}^{e^{56}} \frac{1}{7 u ln ( u )} d u

Take the constant out:

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

= \frac{1}{7} \cdot \int_{e^{7}}^{e^{56}} \frac{1}{u ln ( u )} d u

Apply u-substitution

: \int_{7}^{56} \frac{1}{v} d v

= \frac{1}{7} \cdot \int_{7}^{56} \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{7} [ln ∣ v ∣]_{7}^{56}

Compute the boundaries:

ln (56) - ln (7)

= \frac{1}{7} (ln (56) - ln (7))

Simplify

= \frac{3}{7} ln (2)

\frac{d y}{d x} = e^{- x} + y

\frac{d y}{d x} = 3 x^{2} - 7

\frac{d}{d x} (6 x - 3 cos (x))

x, 4 x, - x + 2

\int \frac{1}{25 + 9 x ^{2}} d x

What is the integral from e to e^8 of 1/(xln(x^7)) ?
The integral from e to e^8 of 1/(xln(x^7)) is 3/7 ln(2)