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int((4-sqrt(2r+1)))/(1-2r)dr

$\int \frac{(4-\sqrt{2 r+1})}{1-2 r} d r$

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Find indefinite integrals using the substitution and by parts

Full solution

Q.

\int \frac{(4-\sqrt{2 r+1})}{1-2 r} d r

Identify Integral: Identify the integral to be solved. $\newline$ We are given the integral $I = \int\left(\frac{4 - \sqrt{2r + 1}}{1 - 2r}\right) dr$ . Our goal is to find the antiderivative of this function.
Simplify Integral: Simplify the integral by breaking it into two separate integrals. $\newline$ $I = \int\frac{4}{1 - 2r} \, dr - \int\frac{\sqrt{2r + 1}}{1 - 2r} \, dr$
Solve First Integral: Solve the first integral $\int\left(\frac{4}{1 - 2r}\right) dr$ . Let $u = 1 - 2r$ , then $du = -2 dr$ , or $dr = -\frac{du}{2}$ . Substitute into the integral: $\int\left(\frac{4}{u}\right) \left(-\frac{1}{2}\right) du = -2 \int\left(\frac{1}{u}\right) du = -2 \ln|u| + C_1$ Replace $u$ with $1 - 2r$ : $-2 \ln|1 - 2r| + C_1$
Solve Second Integral: Solve the second integral $\int(\sqrt{2r + 1} / (1 - 2r)) \, dr$ . This integral is more complex and may require a substitution or partial fractions. However, we notice that the derivative of $2r + 1$ is $2$ , which is related to the denominator $1 - 2r$ . This suggests a substitution might work. Let $u = 2r + 1$ , then $du = 2 \, dr$ , or $dr = du/2$ . Substitute into the integral: $\int(\sqrt{u} / (1 - u/2)) \cdot (1/2) \, du$
Simplify Integral: Simplify the integral $\int(\sqrt{u} / (1 - u/2)) \cdot (1/2) \, du$ . We can rewrite the integral as $(1/2) \int(\sqrt{u} / (1 - u/2)) \, du$ . Now, we need to simplify the denominator: $1 - u/2 = (2 - u)/2$ . The integral becomes $(1/2) \int(\sqrt{u} \cdot 2 / (2 - u)) \, du = \int(\sqrt{u} / (2 - u)) \, du$ .
Solve Integral: Solve the integral $\int(\sqrt{u} / (2 - u)) \, du$ . This integral can be solved by a simple substitution or by recognizing it as a standard integral form. However, upon reviewing the previous steps, we realize that there has been a mistake in the substitution process. The denominator should have been $(2/u - 1)$ instead of $(1 - u/2)$ after the substitution. This is a critical error that affects the solution.

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