Evaluate the following integrals. int_0^3frac{2x-1}{x+1}

Evaluate the following integrals. int_0^3frac{2x-1}{x+1}

Question
Integrals
asked 2021-02-12
Evaluate the following integrals.
\(\int_0^3\frac{2x-1}{x+1}\)

Answers (1)

2021-02-13
\(\text{Consider the given integral}\)
\(\int_0^3\frac{2x-1}{x+1}\)
\(\text{Let }x+1=u\Rightarrow dx=du\)
\(\text{Also, when }x=0\Rightarrow u=1\text{ when }x=3\Rightarrow u=4\)
\(\text{So, the integral becomes:}\)
\(\int_1^4\frac{2(u-1)-1}{u}du\)
\(\Rightarrow\int_1^4\frac{2u-3}{u}du\)
\(\Rightarrow\int_1^4\frac{2u}{u}-\frac{3}{u}du\)
\(\Rightarrow\int_1^4 2-\frac{3}{u}du\)
\(=2u-3\ln (u)|_1^4\)
\(=2(4)-3\ln(4)-2(1)+3\ln(1)\)
\(\Rightarrow8-3\ln(4)-2+3(0)\)
\(\Rightarrow6-3\ln(2^2)\)
\(\Rightarrow6-6\ln(2)\)
\(\text{Hence, }\int_0^3\frac{2x-1}{x+1}dx=6-6\ln(2)\)
0

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