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How do you integrate #x/(x+10)#?

Answer 1

Luke Alvarez

#intx/(x+10)dx=x-10ln|x+10|+C#

A simple substitution will do.

#u=x+10 -> x=u-10#

#du=dx#

Rewrite and simplify:

#int(u-10)/udu=intu/udu-10int(du)/u#

Integrate:

#intdu-10int(du)/u=u-10ln|u|+C#

Rewrite in terms of #x,# yielding

#intx/(x+10)dx=x+10-10ln|x+10|+C#

We may absorb the #10# into #C.#

#intx/(x+10)dx=x-10ln|x+10|+C#

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Answer 2

Ezekiel Alexander

To integrate ( \frac{x}{x+10} ), you can use a technique called partial fraction decomposition. After decomposition, integrate each term separately.

The decomposition of ( \frac{x}{x+10} ) is:

[ \frac{x}{x+10} = \frac{A}{x+10} + \frac{B}{x} ]

To solve for ( A ) and ( B ), multiply both sides by the common denominator ( x(x+10) ), then equate the numerators.

Once you have found ( A ) and ( B ), integrate each term separately:

[ \int \frac{x}{x+10} , dx = A \ln|x+10| + B \ln|x| + C ]

Where ( C ) is the constant of integration.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.