"Find the undefined integral. ∫1x(1+ln⁡x)dx" was answered by our community

khi1la2f1qv

Answered question

2021-11-20

Find the undefined integral.

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

Witheyesse47

Beginner2021-11-21Added 14 answers

Step 1
Given indefinite integral

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

.
Apply method substitution to solve the integral.
Put

1 + \ln x = u, t h e n, \frac{1}{x} d x = d u

Step 2
Then, the integral becomes,

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

= \ln u + c

Plug in

1 + \ln x

for u implies,

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

Thus, the value of integral is

\ln (1 + \ln x) + c

Gloria Lusk

Beginner2021-11-22Added 18 answers

Step 1: Use Integration by Substitution.
Let

u = 1 + \ln x, d u = \frac{1}{x} d x

Step 2: Using u and du above, rewrite

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

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

Step 3: The derivative of

\ln x i s \frac{1}{x}

\ln u

Step 4: Substitute

u = 1 + \ln x

back into the original integral.

\ln (1 + \ln x)

Step 5: Add constant.

\ln (1 + \ln x) + C

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