Evaluate Definite integral using substitution

[Guest]

Evaluate Definite Integral Using Substitution (round answer to the nearest thousandth)

4
/ 2x+6/x^2+6x+3
1

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[stapel]

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Eliz.

 

[soroban]

Hello, amy2310!

Evaluate using substitution (round answer to the nearest thousandth)

. . . \(\displaystyle \L \int_1\,^4\frac{2x\,+\,6}{x^2\,+\,6x\,+\,3}\,dx\)

Let \(\displaystyle u\,=\,x^2\,+\,6x\,+\,3\;\;\Rightarrow\;\;du\,=\,(2x\,+\,6)dx\;\;\Rightarrow\;\;dx\,=\,\frac{du}{2x+6}\)

Substitute: .\(\displaystyle \L \int\frac{2x\,+\,6}{u}\cdot\frac{du}{2x\,+\,6} \;=\;\int\frac{du}{u}\;=\;\ln|u|\)

Back-substsitute: .\(\displaystyle \L\ln|2x\,+\,6|\)

Evaluate: .\(\displaystyle \L\ln|2x\,+\,6|\,\bigg]^4_1\)