Which of the following responses is the expression for \( \int \tan ^{4}(x) d x \) ? \( \frac{\tan ^{3}(x)}{3}-\tan (x)-x+C \) None of these responses is correct. \( \frac{\tan ^{3}(x)}{3}+\tan (x)+x+C \) \( \frac{\tan ^{3}(x)}{3}-\tan (x)+x+C \) \( -\frac{\tan ^{3}(x)}{3}+\tan (x)+x+C \)

Calculate the integral \( \int \tan^{4}(x) dx \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \tan^{4}\left(x\right) dx\) - step1: Evaluate the integral: \(\frac{\tan^{4-1}\left(x\right)}{4-1}-\int \tan^{4-2}\left(x\right) dx\) - step2: Subtract the numbers: \(\frac{\tan^{3}\left(x\right)}{4-1}-\int \tan^{4-2}\left(x\right) dx\) - step3: Subtract the numbers: \(\frac{\tan^{3}\left(x\right)}{4-1}-\int \tan^{2}\left(x\right) dx\) - step4: Subtract the numbers: \(\frac{\tan^{3}\left(x\right)}{3}-\int \tan^{2}\left(x\right) dx\) - step5: Simplify the expression: \(\frac{1}{3}\tan^{3}\left(x\right)-\int \tan^{2}\left(x\right) dx\) - step6: Evaluate the integral: \(\frac{1}{3}\tan^{3}\left(x\right)-\left(\tan\left(x\right)-x\right)\) - step7: Remove the parentheses: \(\frac{1}{3}\tan^{3}\left(x\right)-\tan\left(x\right)+x\) - step8: Add the constant of integral C: \(\frac{1}{3}\tan^{3}\left(x\right)-\tan\left(x\right)+x + C, C \in \mathbb{R}\) The expression for \( \int \tan^{4}(x) dx \) is \( \frac{1}{3}\tan^{3}(x) - \tan(x) + x + C \), where \( C \) is a constant. Therefore, the correct response is \( \frac{\tan^{3}(x)}{3} - \tan(x) + x + C \).