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Math  /  Calculus

QuestionIf we perform the appropriate (i.e. helpful) uu-sub for ∫ln⁡(x)xdx\int \frac{\ln (x)}{x} \mathrm{dx}, what does the new integral look like in terms of uu right after performing the substitution? ∫ln⁡(u)udu\int \frac{\ln (u)}{u} d u ∫udu\int u d u none of these ∫ln⁡(u)xdu\int \frac{\ln (u)}{x} d u

Studdy Solution

STEP 1

1. We are given the integral ∫ln⁡(x)x dx\int \frac{\ln(x)}{x} \, \mathrm{dx}.
2. We need to perform a uu-substitution to simplify the integral.
3. The goal is to express the integral in terms of uu.

STEP 2

1. Choose an appropriate substitution for uu.
2. Differentiate the chosen substitution to find dudu.
3. Substitute uu and dudu into the integral.
4. Simplify the integral in terms of uu.

STEP 3

Choose an appropriate substitution. Let u=ln⁡(x) u = \ln(x) .

STEP 4

Differentiate the substitution to find dudu:
dudx=1x \frac{du}{dx} = \frac{1}{x}
Thus,
du=1x dx du = \frac{1}{x} \, dx

STEP 5

Substitute uu and dudu into the integral:
The original integral is:
∫ln⁡(x)x dx \int \frac{\ln(x)}{x} \, \mathrm{dx}
Substituting u=ln⁡(x)u = \ln(x) and du=1x dxdu = \frac{1}{x} \, dx, we get:
∫u du \int u \, du

STEP 6

The new integral in terms of uu is:
∫u du \int u \, du
This matches the option ∫u du\int u \, du.

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