Problem 10 Find the integrals in problems. ... [FREE SOLUTION]

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Applied Calculus

Deborah Hughes-Hallett, Patti Frazer Lock, Andrew M. Gleason

$Math Studyset 91影视 Explanations$ Math

4 Edition

Chapter 7: Problem 10

Find the integrals in problems. Check your answers by differentiation. $$ \int x^{2}\left(1+2 x^{3}\right)^{2} d x $$

Short Answer

Expert verified

The integral $ \int x^{2}(1+2x^{3})^{2} \, dx = \frac{(1 + 2x^3)^3}{18} + C $.

Step by step solution

Identify the Integral Type

Notice that the integral is of the form $ \int x^2 (1 + 2x^3)^2 \, dx $. This suggests that we can use substitution to simplify the integration.

Choose a Substitution

Let $ u = 1 + 2x^3 $. Then the derivative $ du/dx = 6x^2 $, so $ du = 6x^2 \, dx $. Therefore, $ x^2 \, dx = \frac{1}{6} \, du $.

Substitute into the Integral

Substitute $ u $ and $ du $ into the integral:\[\int x^2 (1 + 2x^3)^2 \, dx = \int \frac{1}{6} u^2 \, du\]

Integrate with Respect to u

Integrate $ \int \frac{1}{6} u^2 \, du $ using the power rule: \[\int \frac{1}{6} u^2 \, du = \frac{1}{6} \cdot \frac{u^3}{3} + C = \frac{u^3}{18} + C\]

Substitute Back to Original Variable

Replace $ u $ with $ 1 + 2x^3 $ to express the integral in terms of $ x $: \[\frac{(1 + 2x^3)^3}{18} + C\]

Differentiate to Verify Solution

Differentiate $ \frac{(1 + 2x^3)^3}{18} + C $ with respect to $ x $ to verify:First calculate the derivative of $ (1 + 2x^3)^3 $, which is $ 3(1 + 2x^3)^2 \cdot 6x^2 $. Hence, the derivative is:\[\frac{1}{18} \cdot 18x^2(1 + 2x^3)^2 = x^2(1 + 2x^3)^2\]This matches the original integrand, confirming the solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution Method

The substitution method is a powerful technique used in integral calculus to simplify complex integrals. By substituting part of the integrand with a new variable, we can transform the integral into an easier form.

In the given problem, we have the integral $ \int x^2 (1 + 2x^3)^2 \, dx $. The expression $(1 + 2x^3)$ is suitable for substitution because its derivative $6x^2$ appears in the integrand.

First, set $ u = 1 + 2x^3 $. This choice aims to simplify the integration process.
Next, calculate the derivative: $ \frac{du}{dx} = 6x^2 $, giving $ du = 6x^2 \, dx $.
Rewrite the original integral: $ x^2 \, dx = \frac{1}{6} \, du $. This allows the substitution.
Substitute into the integral: $ \int x^2 (1 + 2x^3)^2 \, dx = \int \frac{1}{6} u^2 \, du $.

Substitution converts the difficult integral into a simple polynomial, making it easier to solve.

Power Rule

The power rule is a fundamental principle in integral calculus used to integrate polynomial functions. It states that for any power $ n eq -1 $, the integral of $ x^n $ is $ \frac{x^{n+1}}{n+1} $. This rule simplifies integrating powers of variables.

In our scenario, we need to integrate $ \int \frac{1}{6} u^2 \, du $, which is a straightforward application of the power rule.

The function to integrate is $ u^2 $. Apply the power rule: $ \int u^2 \, du = \frac{u^3}{3} $.
Because there's a constant factor of $ \frac{1}{6} $, the final result becomes $ \frac{1}{6} \times \frac{u^3}{3} = \frac{u^3}{18} + C $.
Here, $ C $ represents the constant of integration, accounting for any constant value that could be added.

Thus, using the power rule, the integral of the simplified form $ \frac{1}{6} u^2 $ is attained with ease.

Verification by Differentiation

Verification by differentiation is a technique for checking the correctness of an integral solution. By differentiating the result and obtaining the original integrand, we confirm the integral's accuracy.

To verify our solution, differentiate $ \frac{(1 + 2x^3)^3}{18} + C $ with respect to $ x $.

Start by differentiating $ (1 + 2x^3)^3 $ using the chain rule: $ 3(1 + 2x^3)^2 \cdot 6x^2 $. This yields $ 18x^2(1 + 2x^3)^2 $.
The prefactor $ \frac{1}{18} $ simplifies the derivative to $ x^2(1 + 2x^3)^2 $.
Notice that this is exactly the original integrand, confirming our integration was performed correctly.

Verification by differentiation ensures the solution is sound, providing confidence in the integration process.

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91影视

Find the integrals in problems. Check your answers by differentiation. $$ \int x^{2}\left(1+2 x^{3}\right)^{2} d x $$

Short Answer

Step by step solution

Identify the Integral Type

Choose a Substitution

Substitute into the Integral

Integrate with Respect to u

Substitute Back to Original Variable

Differentiate to Verify Solution

Key Concepts

Substitution Method

Power Rule

Verification by Differentiation

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