Integrate ln((5)/(12)a) da

Set u = (5/12)a: We are asked to integrate the natural logarithm function ln((5/12)a) with respect to a. The integral of ln(x) with respect to x is x*ln(x) - x + C, where C is the constant of integration. We will use this formula to integrate ln((5/12)a) with respect to a.Substitute u and da: First, let's set u = (5/12)a. This means that du/da = 5/12, and thus da = (12/5)du. We will substitute u for (5/12)a and da for (12/5)du in the integral.Apply integration formula: The integral becomes ∫ln(u) * (12/5)du. We can pull the constant (12/5) outside the integral, so it becomes (12/5) * ∫ln(u) du.Simplify the expression: Now we apply the integration formula for ln(u), which is u*ln(u) - u. The integral of ln(u) with respect to u is u*ln(u) - u + C.Final answer: Substituting back in for u, we get (12/5) * ((5/12)a*ln((5/12)a) - (5/12)a) + C.Simplify the expression by multiplying through by (12/5). This gives us a*ln((5/12)a) - a + C', where C' is a new constant of integration that absorbs the (12/5)*(5/12) factor.The final answer is a*ln((5/12)a) - a + C', where C' is the constant of integration.

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Integrate $\newline$ $\ln \left(\frac{5}{12} a\right) \text { da }$

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Q. Integrate

\newline

\ln \left(\frac{5}{12} a\right) \text { da }

Set $u = \frac{5}{12}a$ : We are asked to integrate the natural logarithm function $\ln\left(\frac{5}{12}a\right)$ with respect to $a$ . The integral of $\ln(x)$ with respect to $x$ is $x\cdot\ln(x) - x + C$ , where $C$ is the constant of integration. We will use this formula to integrate $\ln\left(\frac{5}{12}a\right)$ with respect to $a$ .
Substitute $u$ and $da$ : First, let's set $u = \frac{5}{12}a$ . This means that $\frac{du}{da} = \frac{5}{12}$ , and thus $da = \frac{12}{5}du$ . We will substitute $u$ for $\frac{5}{12}a$ and $da$ for $\frac{12}{5}du$ in the integral.
Apply integration formula: The integral becomes $\int \ln(u) \cdot \left(\frac{12}{5}\right)du$ . We can pull the constant $\left(\frac{12}{5}\right)$ outside the integral, so it becomes $\left(\frac{12}{5}\right) \cdot \int \ln(u) du$ .
Simplify the expression: Now we apply the integration formula for $\ln(u)$ , which is $u\cdot\ln(u) - u$ . The integral of $\ln(u)$ with respect to $u$ is $u\cdot\ln(u) - u + C$ .
Final answer: Substituting back in for $u$ , we get \frac{$12$}{$5$} \cdot \left(\frac{$5$}{$12$}a\ln\left(\frac{$5$}{$12$}a\right) - \frac{$5$}{$12$}a\right) + C.
Final answer: Substituting back in for \(u, we get $\frac{12}{5} \times \left(\frac{5}{12}a\ln\left(\frac{5}{12}a\right) - \frac{5}{12}a\right) + C$ . Simplify the expression by multiplying through by $\frac{12}{5}$ . This gives us $a\ln\left(\frac{5}{12}a\right) - a + C'$ , where $C'$ is a new constant of integration that absorbs the $\frac{12}{5}\times\frac{5}{12}$ factor.
Final answer: Substituting back in for $u$ , we get $\frac{12}{5} \times \left(\frac{5}{12}a\ln\left(\frac{5}{12}a\right) - \frac{5}{12}a\right) + C$ . Simplify the expression by multiplying through by $\frac{12}{5}$ . This gives us $a\ln\left(\frac{5}{12}a\right) - a + C'$ , where $C'$ is a new constant of integration that absorbs the $\frac{12}{5}\times\frac{5}{12}$ factor. The final answer is $a\ln\left(\frac{5}{12}a\right) - a + C'$ , where $C'$ is the constant of integration.