Integrate 7 cosh 5t

Step 1: Identify the integral. We need to integrate $7 \cosh(5t)$ with respect to $t$. $\int 7 \cosh(5t) \, dt$

Step 2: Use the constant multiple rule for integration. The constant $7$ can be pulled out of the integral. $7 \int \cosh(5t) \, dt$

Step 3: Apply the integration rule for $\cosh(ax)$. The integral of $\cosh(ax)$ is $\frac{1}{a} \sinh(ax) + C$. In this case, $a=5$. $7 \left( \frac{1}{5} \sinh(5t) \right) + C$

Step 4: Simplify the expression. $\frac{7}{5} \sinh(5t) + C$

The final answer is $\boxed{\frac{7}{5} \sinh(5t) + C}$.

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