Evaluate the integral. (Remember to use absolute values where appropriate. Remember the constant of integration.) $\int 5x \sec(x) \tan(x) \, dx$

Neetesh Kumar | December 8, 2024

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This is the solution to Math 132
Assignment: 7.2 Question Number 8
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Step-by-step solution:

We aim to evaluate the integral $\int 5x \sec(x) \tan(x) \, dx$.

Step 1: Factor out the constant

The constant $5$ can be factored out:

$\int 5x \sec(x) \tan(x) \, dx = 5 \int x \sec(x) \tan(x) \, dx$

Step 2: Recognize the derivative of $\sec(x)$

Recall that the derivative of $\sec(x)$ is $\sec(x) \tan(x)$. This allows us to apply integration by parts.

Step 3: Use integration by parts

Integration by parts is given by:

$\int u \, dv = uv - \int v \, du$

Let:

• $u = x$ (so $du = dx$),
• $dv = \sec(x) \tan(x) \, dx$ (so $v = \sec(x)$, as $\int \sec(x) \tan(x) \, dx = \sec(x)$).

Using integration by parts:

$\int x \sec(x) \tan(x) \, dx = x \sec(x) - \int \sec(x) \, dx$

Step 4: Evaluate the remaining integral

The integral of $\sec(x)$ is:

$\int \sec(x) \, dx = \ln|\sec(x) + \tan(x)| + C$

Substitute this result:

$\int x \sec(x) \tan(x) \, dx = x \sec(x) - \ln|\sec(x) + \tan(x)| + C$

Step 5: Multiply by the constant

Now multiply by the constant $5$:

$5 \int x \sec(x) \tan(x) \, dx = 5 \left( x \sec(x) - \ln|\sec(x) + \tan(x)| \right) + C$

Final Answer:

$\int 5x \sec(x) \tan(x) \, dx = \boxed{5x \sec(x) - 5 \ln(|\sec(x) + \tan(x)|) + C}$

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