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Chapter 5: Problem 2

Find the the general antiderivative \(F(x)\) of the function \(f(x)\) given below. Note that you can check your answer by differentiation. \(f(x)=2 x^{3} \sin \left(x^{4}\right)\) antiderivative \(F(x)=\) ___________________________

Short Answer

Expert verified
\(F(x) = -\frac{1}{2} \cos(x^4) + C \)

Step by step solution

01

- Recognize the integral form

Identify the function to integrate: \( f(x) = 2 x^3 \sin(x^4) \)
02

- Substitution method

Use substitution to simplify the integral. Let \( u = x^4 \). Then, \( du = 4x^3 dx \) and \( dx = \frac{du}{4x^3} \).
03

- Rewrite the integral

Substitute \( u \) and \( dx \) into the integral: \( \int 2x^3 \sin(x^4) dx = \int 2x^3 \sin(u) \frac{du}{4x^3} \).
04

- Simplify the integral

Simplify the expression: \( \int 2x^3 \sin(x^4) \frac{du}{4x^3} = \int \frac{1}{2} \sin(u) du \).
05

- Integrate

Carry out the integration: \( \int \frac{1}{2} \sin(u) du = -\frac{1}{2} \cos(u) + C \).
06

- Substitute back

Replace \( u \) with the original expression \( x^4 \): \( F(x) = -\frac{1}{2} \cos(x^4) + C \).
07

- Verification

Verify by differentiation: Differentiate \( F(x) = -\frac{1}{2} \cos(x^4) + C \) to check if it yields \( f(x) = 2 x^3 \sin(x^4) \).

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Key Concepts

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

substitution method in integration
The substitution method is a powerful technique for finding the antiderivative of complex functions. This method simplifies the integration by making a substitution that reduces the integral to a more manageable form. In our example, we have the function: \( f(x) = 2 x^3 \, \text{sin}(x^4) \) To use the substitution method, we need to identify a part of the function that can be substituted. Here, we let \(u = x^4\). The next step is to calculate \(du\), which is the derivative of \(u\) with respect to \(x\):\( du = 4x^3 \, dx \)Rearrange this to solve for \(dx\):\( dx = \frac{du}{4 x^3} \)Now, substitute \(u\) and \(dx\) back into the original integral to simplify it. This substitution helps convert a complex integral into a simpler one, making it easier to integrate. Steps:
  • Identify a substitution \( u \)
  • Calculate \( du \)
  • Rewrite the integral using \( u \) and \( du \)
integral simplification
Once you have made a substitution, the next step is to simplify the integral. Let's continue with our example. We initially have:\( \int 2x^3 \, \text{sin}(x^4) \, dx \)With our earlier substitution \(u = x^4\) and \(dx = \frac{du}{4 x^3}\), we substitute these into the integral:\( \int 2x^3 \, \text{sin}(u) \, \frac{du}{4 x^3} \)Notice that we can now simplify this expression. The \( x^3 \) terms cancel out, simplifying our integral to:\( \int \frac{1}{2} \, \text{sin}(u) \, du \)This is much easier to integrate compared to the original expression. Simplification helps make the integral feasible to evaluate. Here鈥檚 a quick checklist for simplifying integrals:
  • Cancel out common factors
  • Combine any constants
  • Ensure the resulting integral is in a standard form
verification by differentiation
After finding the antiderivative, it鈥檚 essential to verify your solution by differentiating. This ensures that you have indeed found the correct antiderivative. For our example, after integration, we got:\( F(x) = -\frac{1}{2} \, \text{cos}(x^4) + C \)To verify, we differentiate \( F(x) \) with respect to \( x \).\( F'(x) = -\frac{1}{2} \frac{d}{dx} \text{cos}(x^4) \)Using the chain rule, we get:\( F'(x) = -\frac{1}{2} ( \text{sin}(x^4) ) (4 x^3) \)Simplify this to:\( F'(x) = 2 x^3 \, \text{sin}(x^4) \)This is precisely our original function \( f(x) \). Thus, we verified that our antiderivative is correct.Steps for verification:
  • Differentiate the antiderivative
  • Simplify the derivative expression
  • Check if the result matches the original function

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Most popular questions from this chapter

Calculate the integral \(\int \frac{7 x+3}{x^{2}-3 x+2} d x=\) ________________

For each of the following indefinite integrals, determine whether you would use \(u\) -substitution, integration by parts, neither*, or both to evaluate the integral. In each case, write one sentence to explain your reasoning, and include a statement of any substitutions used. (That is, if you decide in a problem to let \(u=e^{3 x}\), you should state that, as well as that \(\left.d u=3 e^{3 x} d x .\right)\) Finally, use your chosen approach to evaluate each integral. (* one of the following problems does not have an elementary antiderivative and you are not expected to actually evaluate this integral; this will correspond with a choice of "neither" among those given.) a. \(\int x^{2} \cos \left(x^{3}\right) d x\) b. \(\int x^{5} \cos \left(x^{3}\right) d x\left(\right.\) Hint: \(\left.x^{5}=x^{2} \cdot x^{3}\right)\) c. \(\int x \ln \left(x^{2}\right) d x\) d. \(\int \sin \left(x^{4}\right) d x\) e. \(\int x^{3} \sin \left(x^{4}\right) d x\) f. \(\int x^{7} \sin \left(x^{4}\right) d x\)

Consider the indefinite integral \(\int \sin ^{3}(x) d x\). a. Explain why the substitution \(u=\sin (x)\) will not work to help evaluate the given integral. b. Recall the Fundamental Trigonometric Identity, which states that \(\sin ^{2}(x)+\cos ^{2}(x)=1\). By observing that \(\sin ^{3}(x)=\sin (x) \cdot \sin ^{2}(x),\) use the Fundamental Trigonometric Identity to rewrite the integrand as the product of \(\sin (x)\) with another function. c. Explain why the substitution \(u=\cos (x)\) now provides a possible way to evaluate the integral in (b). d. Use your work in (a)-(c) to evaluate the indefinite integral \(\int \sin ^{3}(x) d x\). e. Use a similar approach to evaluate \(\int \cos ^{3}(x) d x\).

For each of the following integrals, indicate whether integration by substitution or integration by parts is more appropriate, or if neither method is appropriate. Do not evaluate the integrals. 1\. \(\int x \sin x d x\) 2\. \(\int \frac{x^{2}}{1+x^{3}} d x\) 3\. \(\int x^{2} e^{x^{3}} d x\) 4\. \(\int x^{2} \cos \left(x^{3}\right) d x\) 5\. \(\int \frac{1}{\sqrt{3 x+1}} d x\) (Note that because this is multiple choice, you will not be able to see which parts of the problem you got correct.)

This problem centers on finding antiderivatives for the basic trigonometric functions other than \(\sin (x)\) and \(\cos (x)\). a. Consider the indefinite integral \(\int \tan (x) d x\). By rewriting the integrand as \(\tan (x)=\) \(\frac{\sin (x)}{\cos (x)}\) and identifying an appropriate function-derivative pair, make a \(u\) -substitution and hence evaluate \(\int \tan (x) d x\). b. In a similar way, evaluate \(\int \cot (x) d x\). c. Consider the indefinite integral $$ \int \frac{\sec ^{2}(x)+\sec (x) \tan (x)}{\sec (x)+\tan (x)} d x $$ Evaluate this integral using the substitution \(u=\sec (x)+\tan (x)\). d. Simplify the integrand in (c) by factoring the numerator. What is a far simpler way to write the integrand? e. Combine your work in (c) and (d) to determine \(\int \sec (x) d x\). f. Using \((\mathrm{c})-(\mathrm{e})\) as a guide, evaluate \(\int \csc (x) d x\).
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