Question

Evaluate the integral ∫(9 tan⁵(x)) dx. (Remember to use absolute values where appropriate. Use c for the constant of integration.)

Answer 1

The integral of 9 tan^5(x) dx is solved using the substitution method, yielding the result 1.5 tan^6(x) + C where C is the constant of integration.

Step-by-step explanation:

To evaluate the integral of 9 tan5(x) dx, we use the substitution method by setting a variable (usually u) to an expression inside the integral that will simplify the integration process. We select u = tan(x), which implies that du = sec2(x) dx. The integral becomes:

∫9u5(1/cos2(x)) du

Since sec(x) is the reciprocal of cos(x), we can rewrite this as:

∫9u5du

The result is a simple power integral that evaluates to:

9/6 u6 + C = 1.5 tan6(x) + C

Where C represents the constant of integration.