Question

Find the indefinite integrals. $$\int\left(\frac{3}{t}-\frac{2}{t^{2}}\right) d t$$

   Find the indefinite integrals.
$$\int\left(\frac{3}{t}-\frac{2}{t^{2}}\right) d t$$

Applied Calculus

Applied Calculus

Deborah… 5th Edition

Chapter 6, Problem 58

Instant Answer

Solved by Expert Linh Vu

07/10/2020

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The integral becomes $$\int\left(3t^{-1}-2t^{-2}\right) dt$$ Show more…

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Find the indefinite integrals. $$\int\left(\frac{3}{t}-\frac{2}{t^{2}}\right) d t$$

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Key Concepts

Indefinite Integration

Indefinite integration involves finding the antiderivative of a function, resulting in a family of functions that differ by an arbitrary constant. It represents the reverse process of differentiation and includes an integration constant because the derivative of a constant is zero.

Power Rule for Integration

The power rule is a fundamental technique for integrating functions of the form t^n. It states that for any real number n ? -1, the antiderivative of t^n is t^(n+1)/(n+1) plus a constant. This rule simplifies the process of integrating polynomial and power functions.

Handling the Special Case of n = -1

When integrating t^(-1), the power rule does not apply because it would involve division by zero. Instead, the integral of t^(-1) is given by the natural logarithm, specifically ln|t|. Recognizing when to use the logarithmic integration is key when dealing with rational functions that contain t in the denominator.

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