Evaluate the integrals. ∫1edxx+ln⁡x}

Question

Evaluate the integrals.  ∫1edxx+ln⁡x}

Gennenzip · Accepted Answer

To find the value of given definite integral.
Given information:

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

Integration formula:
1)

\int \frac{1}{t} d t = \ln t + C

Calculation:

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

Let,

1 + \ln x = v

Differentiate,

d (v) = d (1 + \ln x)

d v = (d (1) + d (\ln x))

d v = (0 + \frac{1}{x} d x)

d v = \frac{1}{x} d x

Change the limits for u.
Lower limit.

1 + \ln x = v

1 + \ln 1 = v

1 + 0 = v

Upper limit.

1 + \ln x = v

1 + \ln e = v

1 + 1 = v

v = 2

Substitute these values into given integral.

\int_{x = 1}^{e} \frac{d x}{x (1 + \ln x)} = \int \frac{1}{(1 + \ln x)} \cdot \frac{d x}{x}

Substitute limits

\int_{x = 1}^{e} {\frac{d x}{x (1 + \ln x) = [\ln v]}}_{v = 1}^{2}

\int_{x = 1}^{e} \frac{d x}{x (1 + \ln x} = \ln 2 - \ln 1}

\int_{x = 1}^{e} \frac{d x}{x (1 + \ln x} = \ln 2 - 0}

\int_{x = 1}^{e} \frac{d x}{x (1 + \ln x} = \ln 2}

Accepted Answer

∫(dx)/(x(1+lnx)(2))

Evaluate the integrals. \int_1^e\frac{dx}{x+\ln x)}