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# Derivative and Integral of Logarithm Help

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By — McGraw-Hill Professional
Updated on Aug 31, 2011

## Introduction to Derivative and Integral of Logarithm

We begin by noting these facts:

If a > 0 then

Math Note : As always, we can state these last formulas more generally as

and

Calculate

#### Solution 1

We see that

For the second problem, we apply our general formulation with a = 3, u = cos x to obtain

Similarly,

Integrate

#### Solution 2

For clarity we set φ( x ) = cot x, φ′ ( x ) = −csc 2 x . Then our integral becomes

Resubstituting the expression for φ( x ) now gives that

You Try It : Evaluate ∫(log 6 ( x 3 )/ x ) dx .

You Try It : Calculate the integral

x · 3 x 2 d x .

Our new ideas about arbitrary exponents and bases now allow us to formulate a general result about derivatives of powers:

For any real exponent a we have

#### Example 3

Calculate the derivative of .

#### Solution 3

We have

You Try It : Calculate ( d/dx )5 sin x−x 2. Calculate (d/dx) x .

Find practice problems and solutions for these concepts at: Transcendental Functions Practice Test.

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#### Q:

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