integral from 1 to 25 of 2/x — step-by-step solution

1) Let's compute the integral:

\int_{1}^{25} \frac{2}{x} d x

\int \frac{2}{x} d x = 2 \cdot \int \frac{1}{x} d x

We use

\int \frac{1}{x} d x = \ln | x |

. The absolute value covers both signs of the denominator, on any interval where

x \neq 0

ⓘ The derivative of

2 \cdot \ln | x |

\frac{2}{x}

, so

2 \cdot \ln | x |

is an antiderivative of

\frac{2}{x}

\int \frac{2}{x} d x

2 \cdot \ln | x |

▷

2 \cdot \ln | x |

2) Let's evaluate for the interval

[1, 25]

2 \cdot \ln | 25 | - 2 \cdot \ln | 1 |

ⓘ For strictly positive arguments, a sum of logarithms is the logarithm of the product, a difference that of the quotient, ln(a) + ln(b) = ln(a*b):

2 \cdot \ln | 25 | - 2 \cdot \ln | 1 |

\ln (\frac{{| 25 |}^{2}}{{| 1 |}^{2}})

\ln (\frac{{| 25 |}^{2}}{{| 1 |}^{2}})

ⓘ An absolute value raised to an even power can be dropped:

| 25 |

25

| 1 |

1

\ln (\frac{25^{2}}{1^{2}})

ⓘ An integer power is a repeated product:

25^{2}

25 \cdot 25

625

\ln (625)

ⓘ The logarithm of a power is a multiple of the logarithm, ln(a^n) = n*ln(a):

\ln (625)

\ln (5^{4})

4 \cdot \ln (5)

▶Integral:

4 \cdot \ln (5)

3) The numeric value of:

4 \cdot \ln (5)

equals:

4 \cdot \approx 1.6094

▷

\approx 6.4378