Find the laplace transform of cosht $$ \int_0^{t}\!e^{u}\,\cos hu\,\mathrm{du} $$

Let's find the Laplace transform of the given function:

\[
f(t) = \int_0^{t} e^u \cosh(u) \, du
\]

### Step 1: Solve the integral
The integrand is $ e^u \cosh(u) $, which simplifies as follows:

Recall that $ \cosh(u) = \frac{e^u + e^{-u}}{2} $, so:

\[
e^u \cosh(u) = e^u \left( \frac{e^u + e^{-u}}{2} \right) = \frac{e^{2u} + 1}{2}
\]

Thus, the integral becomes:

\[
f(t) = \int_0^{t} \frac{e^{2u} + 1}{2} \, du
\]

Now, let's evaluate the integral term-by-term:

\[
\int_0^{t} \frac{e^{2u}}{2} \, du = \frac{1}{2} \cdot \frac{e^{2u}}{2} \bigg|_0^t = \frac{e^{2t} - 1}{4}
\]

\[
\int_0^{t} \frac{1}{2} \, du = \frac{t}{2}
\]

Thus, the result of the integral is:

\[
f(t) = \frac{e^{2t} - 1}{4} + \frac{t}{2}
\]

### Step 2: Find the Laplace transform of $ f(t) $
Now, we take the Laplace transform of $ f(t) = \frac{e^{2t} - 1}{4} + \frac{t}{2} $.

#### Laplace transform of $ \frac{e^{2t} - 1}{4} $:
The Laplace transform of $ e^{at} $ is $ \frac{1}{s - a} $, so:

\[
\mathcal{L}\left\{ e^{2t} \right\} = \frac{1}{s - 2}
\]

The Laplace transform of a constant (in this case, -1) is $ \frac{1}{s} $, so:

\[
\mathcal{L}\left\{ \frac{e^{2t} - 1}{4} \right\} = \frac{1}{4} \left( \frac{1}{s - 2} - \frac{1}{s} \right)
\]

#### Laplace transform of $ \frac{t}{2} $:
The Laplace transform of $ t $ is $ \frac{1}{s^2} $, so:

\[
\mathcal{L}\left\{ \frac{t}{2} \right\} = \frac{1}{2s^2}
\]

### Step 3: Combine the results
Now, combine all the Laplace transforms:

\[
\mathcal{L}\{ f(t) \} = \frac{1}{4} \left( \frac{1}{s - 2} - \frac{1}{s} \right) + \frac{1}{2s^2}
\]

Thus, the Laplace transform of $ \int_0^{t} e^u \cosh(u) \, du $ is:

\[
\boxed{\mathcal{L}\{ f(t) \} = \frac{1}{4} \left( \frac{1}{s - 2} - \frac{1}{s} \right) + \frac{1}{2s^2}}
\]

Would you like more details or have any further questions?

Here are 5 related questions:
1. What is the Laplace transform of $ \cosh(t) $?
2. How would you find the Laplace transform of $ e^t \sin(t) $?
3. What is the Laplace transform of $ t^n $ for a general $ n $?
4. How do you compute the inverse Laplace transform of a rational function?
5. What are the basic properties of the Laplace transform?

**Tip:** The linearity of the Laplace transform makes it much easier to break down complex functions into simpler terms to transform individually!

Find the Laplace transform of cosht $$ \int_0^{t}\!e^{u}\,\cosh(u)\,\mathrm{du} $$

Learn how to compute the Laplace transform of the integral involving e^u and cosh(u). This problem explores the properties of the Laplace transform and hyperbolic functions, using integral calculus. Suitable for undergraduate mathematics students.

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