Math Problem Statement

Evaluate the integral. (Remember the constant of integration.) 4 tan(x) sec3(x) dx

Solution

To evaluate the integral 4tan(x)sec3(x)dx\int 4 \tan(x) \sec^3(x) \, dx, we can proceed by using a substitution method.

Step 1: Use substitution

Let’s use the substitution u=sec(x)u = \sec(x), which gives:

du=sec(x)tan(x)dxdu = \sec(x) \tan(x) \, dx

We can now rewrite the integral in terms of uu. First, recognize that sec3(x)=u3\sec^3(x) = u^3, and sec(x)tan(x)dx=du\sec(x) \tan(x) \, dx = du, so the integral becomes:

4tan(x)sec3(x)dx=4u2du\int 4 \tan(x) \sec^3(x) \, dx = \int 4 u^2 \, du

Step 2: Integrate

Now integrate 4u24 u^2:

4u2du=4u33+C\int 4 u^2 \, du = \frac{4 u^3}{3} + C

Step 3: Substitute back u=sec(x)u = \sec(x)

Now, substitute back u=sec(x)u = \sec(x):

4sec3(x)3+C\frac{4 \sec^3(x)}{3} + C

Final Answer:

Thus, the integral is:

4sec3(x)3+C\frac{4 \sec^3(x)}{3} + C

Would you like more details or have any questions?


Here are 5 related questions to further explore this topic:

  1. What are the other common techniques to solve integrals involving trigonometric functions?
  2. How would the solution change if the integral were tan(x)sec2(x)dx\int \tan(x) \sec^2(x) \, dx?
  3. What is the geometric interpretation of sec(x)\sec(x) in relation to the unit circle?
  4. How would you approach an integral involving tan2(x)\tan^2(x)?
  5. Can we use integration by parts to solve this type of trigonometric integral?

Tip: In integrals involving powers of secant and tangent, substitution with u=sec(x)u = \sec(x) or u=tan(x)u = \tan(x) can often simplify the expression.

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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)