Evaluate the integral: $\int e^{-5t} \cos(-5t) \, dt$

Neetesh Kumar | November 11, 2024

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This is the solution to DHW Calculus
Assignment: 3 Question Number 16
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Step-by-step solution:

Step 1: Simplify the integrand

We can use the property of cosine that $\cos(-x) = \cos(x)$, since cosine is an even function. Therefore:

$\cos(-5t) = \cos(5t)$

Thus, the integral becomes:

$\int e^{-5t} \cos(5t) \, dt$

Step 2: Use the standard formula for integrals of the form $\int e^{at} \cos(bt) \, dt$:

The formula to integrate this type of expression is:

$\int e^{at} \cos(bt) \, dt = \frac{e^{at}}{a^2 + b^2} \left( a \cos(bt) + b \sin(bt) \right) + C$

For our case, we have $a = -5$ and $b = 5$.

Applying the formula:

$\int e^{-5t} \cos(5t) \, dt = \frac{e^{-5t}}{(-5)^2 + 5^2} \left( (-5) \cos(5t) + 5 \sin(5t) \right) + C$

Step 3: Simplify the constants

First, calculate $(-5)^2 + 5^2$:

$(-5)^2 + 5^2 = 25 + 25 = 50$

Thus, the integral becomes:

$\frac{e^{-5t}}{50} \left( -5 \cos(5t) + 5 \sin(5t) \right) + C$

Final Answer:

The final result of the integral is:

$\boxed{\frac{e^{-5t}}{50} \left( -5 \cos(5t) + 5 \sin(5t) \right) + C}$

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