https://r2.erweima.ai/mathbot/image/1726323941471nba9ygm2.jpg

The image contains a few integration problems related to trigonometric functions. I'll walk through the steps visible in the image.

### Problem VI: \(\int \tan^2 x \, dx\)
1. Begin by rewriting \(\tan^2 x\) using a known trigonometric identity:
   \[
   \tan^2 x = \sec^2 x - 1
   \]
   So, the integral becomes:
   \[
   \int \tan^2 x \, dx = \int (\sec^2 x - 1) \, dx
   \]
2. Separate the integrals:
   \[
   = \int \sec^2 x \, dx - \int 1 \, dx
   \]
3. Integrate each part:
   - \(\int \sec^2 x \, dx = \tan x + C\)
   - \(\int 1 \, dx = x\)
   
   Therefore, the solution is:
   \[
   \int \tan^2 x \, dx = \tan x - x + C
   \]

### Problem VII: \(\int \sin^4 x \, dx\)
1. Use the identity for \(\sin^4 x\):
   \[
   \sin^4 x = \left( \frac{1 - \cos 2x}{2} \right)^2
   \]
   So the integral becomes:
   \[
   \int \sin^4 x \, dx = \int \left( \frac{1 - \cos 2x}{2} \right)^2 \, dx
   \]
2. Expand the square:
   \[
   = \frac{1}{4} \int (1 - 2 \cos 2x + \cos^2 2x) \, dx
   \]
3. Break it into three integrals:
   \[
   = \frac{1}{4} \left( \int 1 \, dx - 2 \int \cos 2x \, dx + \int \cos^2 2x \, dx \right)
   \]
4. Solve each integral:
   - \(\int 1 \, dx = x\)
   - \(\int \cos 2x \, dx = \frac{1}{2} \sin 2x\)
   - To integrate \(\cos^2 2x\), use the identity:
     \[
     \cos^2 2x = \frac{1 + \cos 4x}{2}
     \]
     Therefore:
     \[
     \int \cos^2 2x \, dx = \frac{1}{2} \int (1 + \cos 4x) \, dx = \frac{x}{2} + \frac{1}{8} \sin 4x
     \]
5. Putting everything together:
   \[
   = \frac{1}{4} \left( x - \sin 2x + \frac{x}{2} + \frac{1}{8} \sin 4x \right)
   \]
   Simplifying:
   \[
   = \frac{x}{4} - \frac{1}{2} \sin 2x + \frac{x}{8} + \frac{1}{32} \sin 4x + C
   \]

---

Would you like any further details or clarification?

Here are 5 related questions you could explore:
1. How would you integrate \(\tan^4 x\)?
2. What are the steps to integrate \(\cos^4 x\)?
3. How do power-reduction formulas simplify higher powers of sine and cosine in integrals?
4. How would you integrate \(\sin^2 x \cos^2 x\)?
5. How does the identity for \(\sin^4 x\) generalize to other powers of sine?

**Tip:** When dealing with even powers of sine and cosine, using double-angle identities often simplifies the integral considerably.

Integration of trigonometric functions including \(\int \tan^2 x \, dx\) and \(\int \sin^4 x \, dx\)

A step-by-step guide to integrating trigonometric functions, including \(\int \tan^2 x\) and \(\int \sin^4 x\). This problem uses double-angle and power-reduction formulas to simplify the integrals. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation