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

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

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

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