91Ó°ÊÓ

The technique of **u-substitution** is a strategy used in calculus to simplify the integration process. In the context of integrals, some expressions can be complex or cumbersome to integrate directly. By substituting a part of the integrand with a new variable, we can transform the integral into a simpler form.

For the given problem, we have the integral \( \int x^{5} \sqrt{x^{3}+1} \, dx \). The expression \( x^3 + 1 \) inside the square root suggests a substitution. We choose \( u = x^3 + 1 \), which simplifies the process:

• **Derivative of \( u \)**: Differentiating \( u = x^3 + 1 \) gives \( du = 3x^2 \, dx \).
• **Solving for \( dx \)**: Rearranging gives \( dx = \frac{du}{3x^2} \).

This transforms part of the integral into a function of \( u \), simplifying our work. With this substitution, we can rewrite complex expressions involving \( x \) in terms of \( u \), making integration more manageable.

**Indefinite integrals** represent the family of all antiderivatives of a function. When we compute an indefinite integral, we are essentially finding a function whose derivative is the given integrand.

In our problem, after performing the u-substitution, we are tasked with integrating expressions like\[ \frac{1}{3} \int (u^{3/2} - u^{1/2}) \, du \].

The process involves:

• **Antiderivatives**: Finding the antiderivative involves rules of integration, such as the power rule: \( \int u^{n} \, du = \frac{u^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.
• **Simplification**: The resulting antiderivatives from integrating each term \( \int u^{3/2} \, du \) and \( \int u^{1/2} \, du \) are then simplified and combined to form the general solution.

The final result must consider back-substituting \( u \) to its original expression in terms of \( x \) to represent the indefinite integral correctly.

The method of **integration by substitution** is an integral part of solving complex integrals in calculus. By substituting part of the integrand, we attempt to simplify the integration process:

1. **Choose a Substitution**: Identify a part of the integral that can be substituted with a simpler expression \( u \), helping to reduce the complexity.
2. **Modify the Differential**: Replace \( dx \) with an expression in terms of \( du \), using the relationship between variables - derived from differentiating \( u \) with respect to \( x \).
3. **Rewrite the Integral**: Transform all parts of the integral into terms of \( u \) - replacing any instances of \( x \) and \( dx \) completely.

In the example, after choosing \( u = x^3 + 1 \), the integral \( \int x^5 \sqrt{x^3+1} \, dx \) becomes simpler. By substituting and rewriting entirely in \( u \), and cancelling out terms where possible, the integral of functions such as \( (u-1) x^2 \) can be directly solved. This is a versatile technique especially when the inside function's derivative forms part of the integrand, enhancing efficiency in executing indefinite integrals.