https://file.aiquickdraw.com/mathbot/file/1729039807799qt69umhq.jpg

The integral provided is:

\[
\int \sqrt{e^{5x}} \, dx
\]

To solve this using the substitution method (let \( u \)-substitution), let's proceed with the following steps:

### Step 1: Simplify the expression

We can rewrite the integrand \( \sqrt{e^{5x}} \) as:

\[
\sqrt{e^{5x}} = \left(e^{5x}\right)^{1/2} = e^{\frac{5x}{2}}
\]

So, the integral becomes:

\[
\int e^{\frac{5x}{2}} \, dx
\]

### Step 2: Apply substitution

Let:

\[
u = \frac{5x}{2}
\]

Then:

\[
\frac{du}{dx} = \frac{5}{2} \quad \Rightarrow \quad dx = \frac{2}{5} du
\]

### Step 3: Rewrite the integral in terms of \( u \)

Substitute \( u = \frac{5x}{2} \) and \( dx = \frac{2}{5} du \):

\[
\int e^{u} \cdot \frac{2}{5} \, du
\]

Now, factor out the constant \( \frac{2}{5} \):

\[
\frac{2}{5} \int e^{u} \, du
\]

### Step 4: Integrate

The integral of \( e^u \) is simply \( e^u \):

\[
\frac{2}{5} e^u + C
\]

### Step 5: Substitute \( u \) back in terms of \( x \)

Recall that \( u = \frac{5x}{2} \), so:

\[
\frac{2}{5} e^{\frac{5x}{2}} + C
\]

Thus, the solution to the integral is:

\[
\frac{2}{5} e^{\frac{5x}{2}} + C
\]

---

Would you like further details on any step or have any other questions?

Here are 5 questions you might consider based on this:

1. What happens if the exponent is more complex, such as \( e^{5x^2} \)?
2. How can we apply substitution in more complicated integrals involving trigonometric functions?
3. What is the difference between \( u \)-substitution and integration by parts?
4. Can this method be used if the function inside the square root was more complex, like \( \sqrt{e^{5x} + x} \)?
5. How would you evaluate a definite integral for this same problem, say from \( x = 0 \) to \( x = 1 \)?

**Tip:** Always remember to simplify the integrand first, if possible, to make substitution easier.

∫√(e^(5x)) dx using let u; let du method

Learn how to evaluate the integral ∫√(e^(5x)) dx using the u-substitution method. This problem covers key calculus concepts such as integration, exponential functions, and substitution techniques. Suitable for high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation