Problem 22 - Calculus

Question

Use the method of substitution to find each of the following indefinite integrals. $$ \int x^{2}\left(x^{3}+5\right)^{9} d x $$

Step-by-Step Solution

Answer

\( \frac{1}{30}(x^3 + 5)^{10} + C \)

Step 1: Choose a Substitution

Select a substitution that will simplify the integral. For the given integral \( \int x^2 (x^3 + 5)^9 \, dx \), choose \( u = x^3 + 5 \). This substitution simplifies the expression inside the parentheses.

Step 2: Differentiate the Substitution

Find the derivative of \( u \) with respect to \( x \). Since \( u = x^3 + 5 \), differentiate it to get \( \frac{du}{dx} = 3x^2 \).

Step 3: Solve for dx

Rearrange the derivative expression to solve for \( dx \). Thus, \( dx = \frac{du}{3x^2} \).

Step 4: Substitute in the Integral

Substitute \( u \) and \( dx \) into the integral. The integral becomes \( \int x^2 u^9 \cdot \frac{du}{3x^2} \). This simplifies to \( \frac{1}{3} \int u^9 \, du \) after canceling \( x^2 \) from the numerator and denominator.

Step 5: Integrate with Respect to u

Integrate \( \frac{1}{3} \int u^9 \, du \). The integral of \( u^9 \) is \( \frac{u^{10}}{10} \). Multiply by \( \frac{1}{3} \) to get \( \frac{1}{30} u^{10} + C \), where \( C \) is the constant of integration.

Step 6: Substitute Back to x

Replace \( u \) back with \( x^3 + 5 \) to obtain the final answer. The final expression is \( \frac{1}{30}(x^3 + 5)^{10} + C \).

Key Concepts

Method of Substitution

The method of substitution is a cornerstone approach in solving indefinite integrals, akin to finding puzzle pieces that fit together perfectly.
This technique involves replacing a part of the integral with a new variable, thereby simplifying the expression.

A good way to start is by identifying a part of the integral that can be expressed as a new variable, typically denoted by \( u \). For instance:

• Choose \( u = x^3 + 5 \) in \( \int x^{2} (x^3 + 5)^{9} \, dx \).
• The key is to replace complicated expressions with something simpler.

After defining \( u \), differentiate it with respect to \( x \) to find \( \frac{du}{dx} \). This step is crucial as it transforms your variable of substitution with respect to \( x \).
Finally, replace the identified part and the differential in the original integral, making the integration process much more straightforward.
Hence, the integral transforms, bringing the entire problem to a more manageable form.

Integration Techniques

Integration techniques are strategies or methods used to solve integrals, especially when they are not straightforward.
Substitution is just one technique among many that can simplify finding the antiderivative of a function.

Once substitution is applied, it often reduces the integral into a form that's much easier to handle with basic integration rules.

• For the given integral \( \int x^2 (x^3 + 5)^9 \, dx \), substitution allows it to become \( \frac{1}{3} \int u^9 \, du \).
• This simplified integral can now be solved by applying basic power rule for integration.

The power rule for integration states when integrating \( u^n \), you increase the power by one and divide by the new power:
\[ \int u^n \, du = \frac{u^{n+1}}{n+1} + C \]
Thus, using this rule, you integrate \( \int u^9 \, du \) to get \( \frac{u^{10}}{10} + C \).
The once complex integral is now reduced to a simple algebraic expression to solve.

Calculus Problem Solving

Calculus problem-solving builds heavily on understanding core techniques and knowing how to apply them.
This involves identifying appropriate methods, like substitution, to bring a complex calculus problem into a simpler, solvable form.

Here's how one might approach it systematically:

• Start by scanning the problem for patterns or structures suggesting substitution.
• Perform necessary algebraic manipulations to align terms for neat substitution equally.
• Integrate deftly using the standard rules after substitution.

The complete process culminates in re-substituting the original variable after integration to render the final answer in terms of the original variable.
In this step-by-step approach, solving \( \int x^2 (x^3 + 5)^9 \, dx \) using substitution strategy gives \( \frac{1}{30}(x^3 + 5)^{10} + C \).
Recognizing patterns and approaching with a suitable strategy helps tackle indefinite integrals effectively, making calculus an easier field to navigate.