Question

\( \int \tan^2\left(5-\frac{x}{2}\right) dx = \)

• A. \( -\frac{1}{2}\tan\left(5-\frac{x}{2}\right) - x + c \)
• B. \( -2\tan\left(5-\frac{x}{2}\right) + c \)
• C. \( \tan\left(5-\frac{x}{2}\right) + c \)
• D. \( -2\tan\left(5-\frac{x}{2}\right) - x + c \)

Hint:

Always use \( \tan^2\theta = \sec^2\theta - 1 \) to simplify integration involving \( \tan^2\theta \).

Answer 1

The Correct Option is D

Step 1: Use identity for \( \tan^2\theta \).
\[ \tan^2\theta = \sec^2\theta - 1. \]
So,
\[ \int \tan^2\left(5-\frac{x}{2}\right) dx = \int \left[\sec^2\left(5-\frac{x}{2}\right) - 1\right] dx. \]

Step 2: Split the integral.
\[ = \int \sec^2\left(5-\frac{x}{2}\right) dx - \int dx. \]

Step 3: Substitute \( u = 5 - \frac{x}{2} \).
\[ u = 5 - \frac{x}{2}, \quad \frac{du}{dx} = -\frac{1}{2}. \]
\[ dx = -2\,du. \]

Step 4: Transform the integral.
\[ \int \sec^2\left(5-\frac{x}{2}\right) dx = \int \sec^2(u)(-2\,du). \]
\[ = -2 \int \sec^2(u)\,du. \]

Step 5: Integrate.
\[ \int \sec^2(u)\,du = \tan(u). \]
So,
\[ -2 \tan(u). \]

Step 6: Substitute back.
\[ -2\tan\left(5-\frac{x}{2}\right). \]
Now include the second part:
\[ \int dx = x. \]
So total integral becomes:
\[ -2\tan\left(5-\frac{x}{2}\right) - x + C. \]

Step 7: Final conclusion.
Thus, the required integral is:
\[ \boxed{-2\tan\left(5-\frac{x}{2}\right) - x + C}. \]