Problem 598 Integrate: \(\int\left(x^{2}+4\r... [FREE SOLUTION]

Chapter 23: Problem 598

Integrate: \(\int\left(x^{2}+4\right)^{5} 2 x d x\)

Short Answer

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The short answer to the question 'Integrate: \( \int\left(x^{2}+4\right)^{5}2x dx \)' is \( \frac{1}{6}(x^{2}+4)^{6}+C \), where \( C \) is the constant of integration. The solution involved the application of the chain rule, reverse chain rule (u-substitution), and the power rule for integrals.

Step by step solution

- Setup and substitution

The first step is to identify our \( g(x) \) and \( f(x) \) functions. The function \( g(x) \) is identified as \( x^{2}+4 \) and \( f(x) = x^5 \). So, now, we will set \( u=x^{2}+4 \), and thus \( du = 2xdx \). This \( 2xdx \) matches exactly with the \( 2xdx \) multiplier in the integral. Then we can convert the integral into terms of \( u \) instead of \( x \) as \( \int u^{5} du \).

- Integral Calculation

The integral of \( u^{5} \) with respect to \( u \) is a straightforward application of the power rule for integrals, which states \( \int u^n du = \frac{1}{n+1}u^{n+1}+C \). When we apply this, we get: \( \frac{1}{6}u^{6}+C \).

- Substitute back for \( x \)

This last step undoes the substitution we did at the start. We replace \( u \) with \( x^{2} + 4 \) to get the final answer. So, \( \int \left(x^{2}+4\right)^{5}2xdx = \frac{1}{6}(x^{2}+4)^{6}+C \), where \( C \) is the constant of integration. So, we successfully solved this complex-looking integral by recognizing it as an application of the chain rule and using a clever substitution, followed by simple application of the power rule for integrals and substituting back for \( x \).

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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 fundamental technique in integral calculus that simplifies solutions. This method helps us solve integrals by changing the variable of integration to make it easier to handle. In the original exercise, notice how we cleverly replaced the expression \( x^2 + 4 \) with \( u \).
This substitution turns our integral into a simpler form, \( \int u^5 \, du \), which is much easier to integrate.

The steps to using the substitution method include:

Identify the inner function \( g(x) \), which will be replaced by \( u \).
Find \( du \), the derivative of \( g(x) \) times \( dx \), ensuring it matches a part of the integrand.
Rewrite the integral in terms of \( u \).
After solving the integral, replace \( u \) with the original expression to revert back to the variable \( x \).

Using substitution simplifies complex integrals, paving the way for easier and clearer solutions.

Power Rule for Integrals

The power rule for integrals is a convenient and widely-used tool in calculus. It states that if you have an integral of the form \( \int u^n \, du \), you can integrate it to become \( \frac{1}{n+1} u^{n+1} + C \), where \( C \) is the constant of integration.
This rule makes finding antiderivatives of simple powers straightforward.

In the original problem, after substituting, we were left with the expression \( \int u^5 \, du \). Applying the power rule here meant:

Raising the power of \( u \) by one, changing 5 to 6.
Dividing by the new power, resulting in \( \frac{1}{6} u^6 \).
Adding the constant of integration \( C \).

The power rule is extremely useful for dealing with polynomials, streamlining the process of integration.

Chain Rule in Calculus

While the original exercise primarily involves integration, understanding the chain rule in calculus assists in grasping why substitution works. The chain rule in differentiation states that if you have a composite function, the derivative is \( f'(g(x)) \times g'(x) \).
In integration, this concept explains why substitution is valid as the reverse of the chain rule.

When we differentiate, we deal with the rate of change of nested functions. Substitution cleverly employs this concept in reverse by:

Recognizing the pattern of a composite function and its derivative in the integrand.
Switching variables to simplify the integral into a straightforward format.
After integrating, reversing the substitution aligns with how functions un-nest in differentiation.

Understanding the chain rule deepens your appreciation for how substitution and integration techniques tie back into fundamental calculus principles.

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91影视

Integrate: \(\int\left(x^{2}+4\right)^{5} 2 x d x\)

Short Answer

Step by step solution

- Setup and substitution

- Integral Calculation

- Substitute back for \( x \)

Key Concepts

Substitution Method

Power Rule for Integrals

Chain Rule in Calculus

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