https://r2.erweima.ai/mathbot/image/17265164112900cutjl8g.jpg

Let's solve both integrals one by one.

### 3. Integral using Tabular Integration
\[
\int -4x^4 e^{-3x} \, dx
\]

**Step-by-step Tabular Integration:**

Tabular integration is helpful when we have a product of functions that can be differentiated multiple times (like polynomials) and functions that can be integrated repeatedly (like exponentials).

1. **Choose functions**:
   - Let \( u = -4x^4 \) (we'll differentiate this).
   - Let \( dv = e^{-3x} \, dx \) (we'll integrate this).

2. **Create the table**:
   - Differentiate \( u \) and integrate \( dv \) as follows:

\[
\begin{array}{|c|c|}
\hline
\text{Derivative of } u = -4x^4 & \text{Integral of } e^{-3x} \\
\hline
-4x^4 & -\frac{1}{3}e^{-3x} \\
-16x^3 & \frac{1}{9}e^{-3x} \\
-48x^2 & -\frac{1}{27}e^{-3x} \\
-96x & \frac{1}{81}e^{-3x} \\
-96 & -\frac{1}{243}e^{-3x} \\
0 & \frac{1}{729}e^{-3x} \\
\hline
\end{array}
\]

3. **Construct the result**:
   Using the tabular method (alternating signs), the integral becomes:

\[
\int -4x^4 e^{-3x} \, dx = -4x^4 \left( -\frac{1}{3}e^{-3x} \right) + (-16x^3)\left(\frac{1}{9}e^{-3x}\right) + \ldots
\]

Simplifying and collecting terms:

\[
= \frac{4}{3}x^4 e^{-3x} - \frac{16}{9}x^3 e^{-3x} + \frac{48}{27}x^2 e^{-3x} - \frac{96}{81}x e^{-3x} + \frac{96}{243}e^{-3x} + C
\]

\[
= \frac{4}{3}x^4 e^{-3x} - \frac{16}{9}x^3 e^{-3x} + \frac{16}{9}x^2 e^{-3x} - \frac{32}{27}x e^{-3x} + \frac{32}{81}e^{-3x} + C
\]

### 4. Integral using Trigonometric Identities
\[
\int \frac{\sin^5(x)}{\cos^2(x)} \, dx
\]

**Step-by-step Solution:**

1. **Rewrite using trigonometric identities**:
   Recall that:
   \[
   \frac{\sin^5(x)}{\cos^2(x)} = \sin^5(x) \cdot \sec^2(x)
   \]
   Now, let’s rewrite \( \sin^5(x) \) as \( \sin^4(x)\sin(x) \) and use the identity \( \sin^2(x) = 1 - \cos^2(x) \):
   \[
   \sin^5(x) = (\sin^2(x))^2 \sin(x) = (1 - \cos^2(x))^2 \sin(x)
   \]
   Therefore:
   \[
   \int (1 - \cos^2(x))^2 \sin(x) \cdot \sec^2(x) \, dx
   \]

2. **Substitute \( u = \cos(x) \)**:
   - \( du = -\sin(x) dx \).
   The integral becomes:
   \[
   - \int (1 - u^2)^2 \cdot \frac{1}{u^2} \, du
   \]
   Simplifying the expression:
   \[
   - \int \frac{(1 - u^2)^2}{u^2} \, du
   \]
   Expanding the numerator:
   \[
   - \int \frac{1 - 2u^2 + u^4}{u^2} \, du = - \int \left( \frac{1}{u^2} - 2 + u^2 \right) \, du
   \]
   Now integrate each term:
   \[
   - \left( -\frac{1}{u} - 2u + \frac{u^3}{3} \right) + C
   \]

3. **Substitute back \( u = \cos(x) \)**:
   \[
   \frac{1}{\cos(x)} + 2\cos(x) - \frac{\cos^3(x)}{3} + C
   \]

Thus, the final result is:
\[
\sec(x) + 2\cos(x) - \frac{\cos^3(x)}{3} + C
\]

---

Would you like more details or further clarification on these steps? 

Here are 5 related questions to deepen your understanding:

1. How does the tabular method simplify integration by parts?
2. What are some common functions suitable for tabular integration?
3. How can we apply different trigonometric identities to simplify integrals?
4. Why is the substitution method useful for trigonometric integrals?
5. How would the integral change if the power of sine were different?

**Tip**: When solving integrals, identifying patterns like products of polynomials and exponentials (for tabular method) or trigonometric identities can save significant time.

Calcule la siguiente integral, empleando integración tabular: ∫-4x^4 e^{-3x} dx; Calcule la siguiente integral, empleando integración de funciones trigonométricas: ∫sen^5(x)/cos^2(x) dx

Learn how to solve integrals using tabular integration for ∫-4x^4 e^{-3x} dx and integration of trigonometric functions for ∫sen^5(x)/cos^2(x) dx. This guide covers step-by-step solutions including key calculus techniques such as substitution and applying trigonometric identities. Suitable for advanced calculus 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