Final answer:
To evaluate the integral ∫(Sec X Tan X – 6x⁵)Dx, we can use integration techniques. For the first integral, we use substitution method with u = Sec X + Tan X. For the second integral, we use the power rule of integration. Combining the results, the final answer is (1/2)(Sec X + Tan X)² + x⁶ + C.
Step-by-step explanation:
To evaluate the integral ∫(Sec X Tan X – 6x⁵)Dx, we can use integration techniques. First, let's break down the integral into two separate integrals: ∫(Sec X Tan X)Dx and ∫(6x⁵)Dx.
For the first integral, we can use a substitution method. Let u = Sec X + Tan X. Taking the derivative of u with respect to x, we have du = (Sec X Tan X + Sec² X)dx. Rearranging this equation, we get dx = du / (Sec X Tan X + Sec² X).
Substituting the values into the integral, we have ∫(Sec X Tan X)Dx = ∫(u) du = (1/2)u² + C.
Now, for the second integral, we can simply use the power rule of integration. Integrating 6x⁵ with respect to x, we get (6/6)x⁶ + C = x⁶ + C.
Combining the results from both integrals, we have ∫(Sec X Tan X – 6x⁵)Dx = (1/2)(Sec X + Tan X)² + x⁶ + C.