Question

Can someone help me with this problem

Answer 1

By applying integration rules and logarithm properties, the indefinite integral of the expression 9/(6 · eˣ + 4) is equal to the logarithmic expression $\frac{9}{4}\cdot \ln \left|\frac{e^{x}}{6\cdot e^{x}+4} \right|$.

How to define the indefinite integral

In this question we must apply algebraic substitution and partial fractions to find the indefinite integral:

$\int {\frac{9}{6\cdot e^{x}+4} } \, dx$

$\int {\frac{9\cdot e^{x}}{(6\cdot e^{x}+4)\cdot e^{x}} } \, dx$

$9\int {\frac{du}{(6\cdot u + 4)\cdot u} }$

Partial fraction

$-\frac{27}{2} \int {\frac{du}{6\cdot u + 4} } +\frac{9}{4} \int {\frac{du}{u} }$

$-\frac{9}{4}\cdot \ln (6\cdot u + 4) + \frac{9}{4}\cdot \ln u$

$\frac{9}{4}\cdot \ln \left|\frac{u}{6\cdot u + 4} \right|$

$\frac{9}{4}\cdot \ln \left|\frac{e^{x}}{6\cdot e^{x}+4} \right|$

By applying integration rules and logarithm properties, the indefinite integral of the expression 9/(6 · eˣ + 4) is equal to the logarithmic expression $\frac{9}{4}\cdot \ln \left|\frac{e^{x}}{6\cdot e^{x}+4} \right|$.

To learn more on indefinite integrals: https://brainly.com/question/27746481

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