Simplify Integral: Let's simplify the integral: int(5)/(2x+1)dx We can factor out the constant 5: 5 * int(1/(2x+1))dxUse Substitution: Now, let's use a substitution to make it easier: Let u = 2x + 1, then du = 2dx. So, dx = du/2.Substitute and Simplify: Substitute and simplify: 5 * int(1/u * du/2) = 5/2 * int(1/u)duIntegrate with Respect: Integrate with respect to u: 5/2 * ln|u| + CSubstitute Back: Substitute back for x: u = 2x + 1 5/2 * ln|2x + 1| + C

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$\int \frac{5}{2 x+1} d x$

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Calculus

Find indefinite integrals using the substitution

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\int \frac{5}{2 x+1} d x

Simplify Integral: Let's simplify the integral: $\int \frac{5}{2x+1}\,dx$ We can factor out the constant $5$ : $5 \times \int \frac{1}{2x+1}\,dx$
Use Substitution: Now, let's use a substitution to make it easier: $\newline$ Let $u = 2x + 1$ , then $du = 2dx$ . $\newline$ So, $dx = \frac{du}{2}$ .
Substitute and Simplify: Substitute and simplify: $\newline$ $5 \times \int(\frac{1}{u} \cdot \frac{du}{2})$ $\newline$ = $\frac{5}{2} \times \int(\frac{1}{u})du$
Integrate with Respect: Integrate with respect to $u$ : $\frac{5}{2} \cdot \ln|u| + C$
Substitute Back: Substitute back for $x$ : $u = 2x + 1$ $\frac{5}{2} \cdot \ln|2x + 1| + C$