Do an appropriate u-substitution to write the integral in terms of u.

3 tan8(x)sec4(x) dx

   	= 
3 tan8(x) sec2(x) sec2(x) dx

To solve the integral \( 3 \tan^8(x) \sec^4(x) \, dx \), we can apply an appropriate substitution. First, let's rewrite the given expression:

\[
3 \tan^8(x) \sec^4(x) \, dx = 3 \tan^8(x) \sec^2(x) \sec^2(x) \, dx
\]

Now, notice that \( \sec^2(x) \, dx \) is the derivative of \( \tan(x) \). This suggests that we can use the following substitution:

Let:
\[
u = \tan(x)
\]
Then:
\[
du = \sec^2(x) \, dx
\]

Now, rewrite the integral in terms of \( u \). Using the substitution \( u = \tan(x) \), the original expression becomes:

\[
3 u^8 \sec^2(x) \, du
\]

Since \( du = \sec^2(x) \, dx \), this simplifies further to:

\[
3 u^8 \, du
\]

Now, the integral becomes:

\[
3 \int u^8 \, du
\]

To solve this, use the power rule of integration:

\[
\int u^n \, du = \frac{u^{n+1}}{n+1}
\]

Applying this to \( u^8 \), we get:

\[
3 \cdot \frac{u^9}{9} = \frac{u^9}{3}
\]

Finally, substitute \( u = \tan(x) \) back in:

\[
\frac{\tan^9(x)}{3} + C
\]

Thus, the integral is:

\[
\boxed{\frac{\tan^9(x)}{3} + C}
\]

Would you like more details on any part of this process?

---

Here are five related questions:
1. How would you solve integrals involving products of trigonometric functions?
2. Can you explain how to differentiate the result \( \frac{\tan^9(x)}{3} \)?
3. What is the integral of \( \sec^2(x) \)? 
4. How do you integrate \( \tan^n(x) \) for different values of \( n \)?
5. Why is substitution a useful method for solving integrals?

**Tip:** When choosing a substitution in integrals, look for patterns that simplify the differential, especially derivatives of trigonometric functions.

Do an appropriate u-substitution to write the integral in terms of u.

3 tan^8(x) sec^4(x) dx

= 3 tan^8(x) sec^2(x) sec^2(x) dx

Learn how to apply u-substitution to solve the integral 3 tan^8(x) sec^4(x) dx. This detailed guide walks through rewriting the integral using trigonometric identities, applying substitution, and integrating using the power rule. Ideal for students in Grades 11-12 or early college calculus courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation