Fundamental Theorem of Calculus Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

medium
Use FTC Part 2 to evaluate ∫1e1t dt\int_1^e \frac{1}{t}\,dt.

Solution

  1. 1
    Antiderivative of 1t\frac{1}{t} is ln⁡∣t∣\ln|t|.
  2. 2
    Apply FTC Part 2: [ln⁡t]1e=ln⁡e−ln⁡1=1−0=1[\ln t]_1^e = \ln e - \ln 1 = 1 - 0 = 1.

Answer

11
The natural logarithm is defined so that ∫1e1t dt=1\int_1^e \frac{1}{t}\,dt = 1. This is actually one way to define ee: it is the number such that the area under 1/t1/t from 1 to ee equals exactly 1.

About Fundamental Theorem of Calculus

The theorem stating that differentiation and integration are inverse operations, linking antiderivatives to definite integrals.

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More Fundamental Theorem of Calculus Examples

Example 1 easy

Let [formula]. Find [formula] using FTC Part 1.

Example 2 hard

Let [formula]. Find [formula].

Example 4 medium

If [formula], find [formula].

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Related Concepts

DerivativeIntegral