\( \left. \begin{array} { l } { \int 11 \tan ^ { 5 } x \sec ^ { 4 } x d x } \\ { \int 11 \tan ^ { 5 } x \sec ^ { 4 } x d x = \square } \end{array} \right. \)

To solve the integral \( \int 11 \tan^5 x \sec^4 x \, dx \), we can use substitution. Notice that if we let \( u = \tan x \), then \( du = \sec^2 x \, dx \). We have \( \sec^4 x = (1 + \tan^2 x) \sec^2 x = (1 + u^2) \, du \). Then, rewriting the integral, we have \( \int 11 u^5 (1 + u^2) \, du \). By expanding this and integrating term by term, we can simplify the integral. Once you calculate it, don’t forget to substitute back in terms of \( x \) to finalize your answer! Integrating leads to: \[ \int 11 (u^5 + u^7) \, du = \frac{11}{6} u^6 + \frac{11}{8} u^8 + C \] Substituting back to \( x \): \[ = \frac{11}{6} \tan^6 x + \frac{11}{8} \tan^8 x + C \] Now you have beautifully solved that integral! What about those common mistakes, right? When tackling integrals involving trigonometric functions, one frequent slip is forgetting to substitute completely. Make sure every part is replaced properly! Another mistake is failing to apply the power rule appropriately on polynomial terms; remember, each exponent gets increased by one, and you divide by the new exponent. Staying mindful of these details will definitely help avoid integration mishaps!