Evaluate the Integral

Problem

(∫_^)(sec(3*x)*tan(3*x)*d(x))

Solution

1. Identify the standard integral form for the product of secant and tangent, which is (∫_^)(sec(u)*tan(u)*d(u))=sec(u)+C

2. Apply a substitution by letting u=3*x

3. Differentiate the substitution to find d(u)=3*d(x) which implies d(x)=1/3*d(u)

4. Substitute the variables into the integral to get (∫_^)(sec(u)*tan(u)⋅1/3*d(u))

5. Factor out the constant 1/3 to get 1/3*(∫_^)(sec(u)*tan(u)*d(u))

6. Integrate the expression with respect to u to obtain 1/3*sec(u)+C

7. Back-substitute u=3*x to express the final result in terms of x

Final Answer

(∫_^)(sec(3*x)*tan(3*x)*d(x))=1/3*sec(3*x)+C

Want more problems? Check here!