Question

The value of `int_-5^-1 1/x dx` is equal to ______.

विकल्प

• −log 5

• x⁶

• log (−5)

• x⁻⁶

Answer 1

The value of `int_-5^-1 1/x dx` is equal to −log 5.

Explanation:

Step 1: Find the Antiderivative

The standard integral of `1/x` is log|x|

It is essential to use the absolute value sign since the natural logarithm is only specified for positive values, and our interval [−5, −1] consists of negative values.

Step 2: Apply the Fundamental Theorem of Calculus

Substitute the upper limit (−1) and the lower limit (−5)

`int_-5^-1 1/x dx = [log |x|]_-5^-1`

= log |−1| − log |−5|

Since log |−1| − log |−5| = 5:

= log(1) − log(5)

The natural log of 1 is always zero (log 1 = 0):

= 0 − log 5

= −log 5