Problem 23 - The Calculus with Analytic Geometry

Question

\(\int \frac{y+3}{(3-y)^{2 / 3}} d y\)

Step-by-Step Solution

Answer

-18(3-y)^{1/3} + \frac{3}{4}(3-y)^{4/3} + C

Step 1: Identify the substitution

Let us use the substitution method. Set \( u = 3 - y \). Then, \( du = -dy \), and \( y = 3 - u \).

Step 2: Rewrite the integral

Rewrite the integral in terms of \( u \). The integral given is \( \int \frac{y+3}{(3-y)^{2 / 3}} dy \). With the substitution \( y = 3 - u \), the expression becomes: \( \int \frac{3-u+3}{u^{2 / 3}} (-du) \). This simplifies to: \( - \int \frac{(6-u)}{u^{2 / 3}} du \).

Step 3: Split the integral

Separate the integral into two easier integrals: \( - \int 6u^{-2 / 3} du + \int u^{1 / 3} du \).

Step 4: Integrate each term

Integrate each term separately: \[-6 \int u^{-2 / 3} du = -6 \left( \frac{u^{1/3}}{1/3} \right) = -18u^{1/3} \] and \[ \int u^{1/3} du = \left( \frac{u^{4/3}}{4/3} \right) = \frac{3}{4}u^{4/3} \].

Step 5: Combine the results

Combine the results of the integrals: \[ -18u^{1/3} + \frac{3}{4}u^{4/3} \].

Step 6: Substitute back for y

Substitute \( u = 3 - y \) back into the expression: \[ -18(3-y)^{1/3} + \frac{3}{4}(3-y)^{4/3} + C \].

Key Concepts

definite integral

A definite integral represents the area under a curve between two specific points. Here, we did not specify any limits of integration, so we worked with an indefinite integral. However, once you understand indefinite integrals, the leap to definite ones becomes easier.
To solve a definite integral using substitution:

• Determine the new limits by substituting the original bounds into the substitution equation.
• Evaluate the integral within these new limits.

For example, if we had limits from 1 to 2 for the original integral, after substituting, we need to find the new bounds corresponding to these values for the new variable.

integration techniques

Integration techniques are methods used to find integrals. One powerful method is integration by substitution. This method simplifies a complex integral into a more manageable form by changing variables.
Here's how to do it:

• Identify part of the integral that you can substitute with a simpler expression.
• Define a new variable, usually denoted as 'u'. For example, we set \( u = 3 - y \).
• Rewrite each part of the integral in terms of 'u' and the corresponding differential \( du \).
• Simplify and integrate.
• Finally, substitute back the original variable.

This method is also known as the 'reverse chain rule' because it essentially undoes the chain rule of differentiation.

change of variables

Change of variables, or substitution, is key in simplifying and solving integrals. Here, we changed variable from 'y' to 'u'.
The process follows these steps:

• Identify the substitution: We set \( u = 3 - y \).
• Update differentials: From \( u = 3 - y \), we get \( du = -dy \).
• Rewrite the integral: Convert all occurrences of 'y' and 'dy' into 'u' and 'du'.
• Integrate using new variable: We separated and integrated \( - \int 6u^{-2 / 3} du + \int u^{1 / 3} du \).
• Re-substitute the original variable: Back-substitute \( u = 3 - y \) into the integrated result.

This method helps explore complicated integrals easily by replacing them with simpler functions.