We need to evaluate the integral:

$\int e^r \left( 9 + 5e^r \right)^{\frac{3}{2}} \, dr$

Substitution

Let: $u = 9 + 5e^r$ Then, differentiating both sides: $du = 5e^r \, dr$ Solving for $e^r \, dr$: $\frac{du}{5} = e^r \, dr$

Rewriting the Integral

Substituting in terms of $u$:

$\int e^r \left( 9 + 5e^r \right)^{\frac{3}{2}} \, dr = \int \left( u^{\frac{3}{2}} \right) \frac{du}{5}$

$= \frac{1}{5} \int u^{\frac{3}{2}} \, du$

Integration

Using the power rule:

$\int u^n \, du = \frac{u^{n+1}}{n+1}$

where $n = \frac{3}{2}$:

$\frac{1}{5} \cdot \frac{u^{\frac{5}{2}}}{\frac{5}{2}}$

$= \frac{1}{5} \cdot \frac{2}{5} u^{\frac{5}{2}}$

$= \frac{2}{25} u^{\frac{5}{2}}$

Substituting Back $u = 9 + 5e^r$

$\frac{2}{25} (9 + 5e^r)^{\frac{5}{2}} + C$

Final Answer:

$\frac{2}{25} (9 + 5e^r)^{\frac{5}{2}} + C$

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Would you like a more detailed breakdown or have any questions?

Related Questions:

1. How do you solve similar integrals with different exponents?
2. What happens if we change the base of the exponent in $e^r$?
3. Can you solve this using integration by parts?
4. How do definite integrals change the approach to this problem?
5. What is the significance of substitution in solving exponent-based integrals?

Tip:

When dealing with exponentials in integration, substitution often simplifies the integral significantly!