Key Concepts and Formulas

• Indefinite Integral: An indefinite integral represents the family of functions whose derivative is the integrand. We denote it as $\int f(x) \, dx = F(x) + C$, where $F'(x) = f(x)$ and $C$ is the constant of integration.
• Trigonometric Identity: $1 + \tan^2 x = \sec^2 x$
• Substitution Method: If $\int f(g(x))g'(x) \, dx$ can be simplified by substituting $u = g(x)$, then $du = g'(x) \, dx$ and the integral becomes $\int f(u) \, du$.

Step-by-Step Solution

Step 1: Write down the given information. We are given $I_n = \int \tan^n x \, dx$, where $n > 1$. We are also given that $I_4 + I_6 = a \tan^5 x + bx^5 + C$, and we need to find the ordered pair $(a, b)$.

Step 2: Express $I_4$ and $I_6$ in terms of integrals. $I_4 = \int \tan^4 x \, dx$ and $I_6 = \int \tan^6 x \, dx$.

Step 3: Substitute $I_4$ and $I_6$ into the given equation and simplify the integral. We have $I_4 + I_6 = \int \tan^4 x \, dx + \int \tan^6 x \, dx = \int (\tan^4 x + \tan^6 x) \, dx$. We can factor out $\tan^4 x$ from the integrand: $\int (\tan^4 x + \tan^6 x) \, dx = \int \tan^4 x (1 + \tan^2 x) \, dx$

Step 4: Use the trigonometric identity to simplify the integrand. Since $1 + \tan^2 x = \sec^2 x$, we have $\int \tan^4 x (1 + \tan^2 x) \, dx = \int \tan^4 x \sec^2 x \, dx$

Step 5: Use the substitution method to evaluate the integral. Let $t = \tan x$. Then, $dt = \sec^2 x \, dx$. Substituting into the integral, we get $\int \tan^4 x \sec^2 x \, dx = \int t^4 \, dt$

Step 6: Evaluate the integral with respect to $t$. $\int t^4 \, dt = \frac{t^5}{5} + C$

Step 7: Substitute back to express the result in terms of $x$. Since $t = \tan x$, we have $\frac{t^5}{5} + C = \frac{\tan^5 x}{5} + C$

Step 8: Compare the result with the given expression $a \tan^5 x + bx^5 + C$. We have $\frac{1}{5} \tan^5 x + C = a \tan^5 x + bx^5 + C$. Comparing the coefficients of $\tan^5 x$ and $x^5$, we get $a = \frac{1}{5}$ and $b = 0$.

Step 9: Write the ordered pair $(a, b)$. The ordered pair is $(a, b) = \left(\frac{1}{5}, 0\right)$.

Common Mistakes & Tips

• Forgetting the constant of integration: Always remember to add the constant of integration $C$ when evaluating indefinite integrals.
• Incorrect trigonometric identities: Ensure you use the correct trigonometric identities to simplify the integrand.
• Choosing the right substitution: Selecting the appropriate substitution is crucial for simplifying the integral. In this case, $t = \tan x$ made the integration straightforward.

Summary

We were given an expression involving integrals of powers of $\tan x$ and asked to find the values of $a$ and $b$ when the expression was simplified to the form $a \tan^5 x + bx^5 + C$. By combining the integrals, using a trigonometric identity, and applying the substitution method, we were able to simplify the integral to $\frac{1}{5} \tan^5 x + C$. Comparing this with the given form, we found that $a = \frac{1}{5}$ and $b = 0$. Therefore, the ordered pair $(a, b)$ is $\left(\frac{1}{5}, 0\right)$.

Final Answer The final answer is \boxed{\left( {{1 \over 5},0} \right)}, which corresponds to option (A).