Math Problem Statement
Evaluate the integral. (Remember the constant of integration.) 4 tan(x) sec3(x) dx
Solution
To evaluate the integral , we can proceed by using a substitution method.
Step 1: Use substitution
Let’s use the substitution , which gives:
We can now rewrite the integral in terms of . First, recognize that , and , so the integral becomes:
Step 2: Integrate
Now integrate :
Step 3: Substitute back
Now, substitute back :
Final Answer:
Thus, the integral is:
Would you like more details or have any questions?
Here are 5 related questions to further explore this topic:
- What are the other common techniques to solve integrals involving trigonometric functions?
- How would the solution change if the integral were ?
- What is the geometric interpretation of in relation to the unit circle?
- How would you approach an integral involving ?
- Can we use integration by parts to solve this type of trigonometric integral?
Tip: In integrals involving powers of secant and tangent, substitution with or can often simplify the expression.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Trigonometric Integrals
Substitution Method
Secant and Tangent Functions
Formulas
∫ 4 tan(x) sec^3(x) dx
Substitution: u = sec(x), du = sec(x) tan(x) dx
Integral of a power of u: ∫ u^n du = (u^(n+1))/(n+1) + C
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
Grades 11-12 (Advanced High School Calculus or Early College)
Related Recommendation
How to Integrate 8 tan(x) sec^3(x): Step-by-Step Solution
Integral of (3 * tan^2(x)) / sec(x) using Trigonometric Substitution
How to Integrate tan^3(4x) Using Trigonometric Identities
Solve the Integral of 3 tan^8(x) sec^4(x) Using U-Substitution
Integral of sec(x) / (sec(x) + tan(x)) - Step-by-Step Solution