Title: Finding Indefinite Integrals---In this tutorial, we will learn how to find indefinite integrals of a given function. Consider the following expression:\[ \int (3x^4 + 4)^5 \cdot 12x^3 \, dx \]We aim to integrate this function with respect to \(x\).**Explanation:**1. Identify the integral expression.2. Recognize if a substitution method can simplify the integration.**Step-by-Step Integration:**1. **Substitution Method:** - Let \( u = 3x^4 + 4 \). - Thus, the differential \( du \) can be found by differentiating \( u \) with respect to \( x \): \[ \frac{du}{dx} = 12x^3 \] or equivalently, \[ du = 12x^3 \, dx \]2. **Rewrite the Integral:** - Substitute \( u \) and \( du \) back into the integral expression: \[ \int (3x^4 + 4)^5 \cdot 12x^3 \, dx = \int u^5 \, du \]3. **Integrate:** - Integrate \( u^5 \) with respect to \( u \): \[ \int u^5 \, du = \frac{u^6}{6} + C \] where \( C \) is the constant of integration.4. **Back-Substitute \( u \):** - Replace \( u \) with the original expression \( 3x^4 + 4 \): \[ \frac{(3x^4 + 4)^6}{6} + C \]5. **Final Answer:** \[ \int (3x^4 + 4)^5 \cdot 12x^3 \, dx = \frac{(3x^4 + 4)^6}{6} + C \]By following these steps, you can solve similar indefinite integral problems using substitution and integration techniques.

Name: Title: Finding Indefinite Integrals---In this tutorial, we will learn
Uploaded: 2024-03-20T07:06:41.000Z
Channel: Bartleby
Description: Title: Finding Indefinite Integrals---In this tutorial, we will learn how to find indefinite integrals of a given function. Consider the following expression:\[ \int (3x^4 + 4)^5 \cdot 12x^3 \, dx \]We aim to integrate this function with respect to