Question

Evaluate the integral: \[ I = \int (\tan^7 x + \tan x) dx. \]

• A. \( \frac{\tan^2 x}{12} (2\tan^4 x - 3\tan^2 x + 6) + C \)
• B. \( \frac{\tan^2 x}{6} - \frac{\tan^5 x}{4} + \frac{\tan^4 x}{2} + C \)
• C. \( \frac{\tan^2 x}{6} (\tan^4 x + 3\tan^2 x + 4) + C \)
• D. \( \frac{\tan x}{12} (\tan^4 x - 3\tan^2 x + 6) + C \)

Hint:

For trigonometric integrals of high powers, use trigonometric identities and reduction formulas to express the function in terms of lower-degree functions.

Answer 1

The Correct Option is A

Step 1: Splitting the Integral We split the given integral into two parts: \[ I = \int \tan^7 x \,dx + \int \tan x \,dx. \] The second integral is straightforward: \[ \int \tan x \,dx = \ln |\sec x| + C. \] Step 2: Expressing \( \tan^7 x \) in Reducible Form Using the identity: \[ \tan^7 x = \tan^3 x \cdot \tan^2 x \cdot \tan^2 x, \] and expressing it in a reducible form, we integrate step by step using substitution techniques. Step 3: Final Integral Using integration techniques, the final answer is: \[ I = \frac{\tan^2 x}{12} (2\tan^4 x - 3\tan^2 x + 6) + C. \]