Final answer:
To evaluate the integral ∫7tan(x)sec³(x)dx, we can simplify it using a trigonometric identity and then make a substitution to make the integral easier to solve. The final result is (tan³(x)/3) + 7tan(x) + C, where C is the constant of integration.
Step-by-step explanation:
To evaluate the integral ∫7tan(x)sec³(x)dx, we can use a trigonometric identity to simplify the integral. The identity is: sec²(x) = tan²(x) + 1. We can rearrange this identity to solve for sec³(x): sec³(x) = sec²(x) * sec(x) = (tan²(x) + 1) * sec(x).
Now, substituting this expression for sec³(x) in the integral, we get: ∫7tan(x)sec³(x)dx = ∫7tan(x)(tan²(x) + 1)sec(x)dx.
At this point, we can make a substitution by letting u = tan(x). Then, du = sec²(x)dx. Substituting these values, the integral becomes: ∫7(u² + 1)du = ∫7u²du + ∫7du = u³/3 + 7u + C = (tan³(x)/3) + 7tan(x) + C, where C is the constant of integration.