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Chapter 7: Problem 15

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

Short Answer

Expert verified
The integral is \( \frac{(x^2 + 3)^3}{6} + C \).

Step by step solution

01

Identify the Method

The given integral \( \int x(x^2 + 3)^2 \, dx \) is suitable for a substitution method. The substitution method simplifies integrals, especially when involving a function and its derivative.
02

Choose the Substitution

Let \( u = x^2 + 3 \). Then, differentiate to find \( du \):\[ du = 2x \, dx \]Solve for \( dx \):\[ dx = \frac{du}{2x} \]
03

Substitute and Simplify the Integral

Substitute \( u \) and \( dx \) in the integral:\[ \int x(u)^2 \, \frac{du}{2x} \] Cancel out \( x \) in the numerator and denominator:\[ \frac{1}{2} \int u^2 \, du \]
04

Integrate

Integrate \( u^2 \) with respect to \( u \):\[ \frac{1}{2} \cdot \frac{u^3}{3} = \frac{u^3}{6} \]
05

Substitute Back the Original Variable

Replace \( u \) with \( x^2 + 3 \):\[ \frac{(x^2 + 3)^3}{6} + C \]Where \( C \) is the constant of integration.
06

Differentiate to Verify

Differentiate \( \frac{(x^2 + 3)^3}{6} + C \) with respect to \( x \):Apply the chain rule:\[ \frac{1}{6} \cdot 3(x^2 + 3)^2 \cdot 2x = x(x^2 + 3)^2 \]This matches the original function, 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.

Integration by Substitution
Integration by substitution is a fundamental technique in integral calculus that is incredibly useful when dealing with complex integrals. The primary goal of this method is to simplify the integration process by transforming a difficult integral into an easier one through a change of variable.
  • When you choose a substitution, you are essentially picking a new variable, usually denoted as \( u \), to replace a part of the integrand (the function being integrated).
  • This new variable, \( u \), should ideally represent the inside of a composite function or a part of the function whose derivative is also present in the integrand.
  • The success of the substitution hinges on correctly identifying this relationship, allowing the integral to be rewritten in terms of \( u \).

    • For example, in the given problem \( \int x(x^2 + 3)^2 \, dx \), a suitable substitution is \( u = x^2 + 3 \). The derivative \( du \) is simply \( 2x \, dx \). By substituting \( u \) and adjusting terms to match \( dx \), the integral simplifies to a more manageable form: \( \frac{1}{2} \int u^2 \, du \). This highlights the power of integration by substitution in breaking down complex problems into simpler ones.
Definite and Indefinite Integrals
Integrals are categorized into two main types: definite integrals and indefinite integrals. Understanding both is crucial for solving and verifying calculus problems.
  • Indefinite integrals: These are integrals without specific limits. They yield a family of functions known as antiderivatives, which include a constant integration \( C \) because differentiation removes constants.
  • Definite integrals: In contrast, these integrals have upper and lower limits. They calculate the area under a curve between two points and result in a specific numerical value, with no arbitrary constant.

    • In our problem, when we perform the integration by substitution, we obtain \( \frac{(x^2 + 3)^3}{6} + C \). This expression represents the antiderivative of the original function, hence an indefinite integral. We add \( C \) to indicate that any constant added to this result will also be a solution to the integral.
    Knowing the differences between definite and indefinite integrals helps students apply correct solving techniques and verifies the outcomes appropriately.
Verification by Differentiation
Verification by differentiation is a powerful technique to confirm the correctness of integrated results. After calculating an integral, differentiating the result should yield the original integrand.
  • If differentiating the integrated function leads back to the original integrand, the solution is verified as correct.
  • This technique relies on the fundamental theorem of calculus, establishing that integration and differentiation are inverse operations.
  • To verify, apply the chain rule correctly during differentiation whenever a composite function is involved.

    • In the exercise problem, after integrating and arriving at \( \frac{(x^2 + 3)^3}{6} + C \), we differentiate this expression. By using the chain rule, we differentiate inside the power \((x^2 + 3)^3\) first and multiply by the derivative of the inside \(2x\). Finally, we simplify to confirm \( x(x^2 + 3)^2 \) matches the original function under the integral. This solidifies the correctness of your integration work through differentiation.

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Most popular questions from this chapter

Decide whether the statements are true or false. Give an explanation for your answer.. The midpoint rule approximation to \(\int_{0}^{1}\left(y^{2}-1\right) d y\) is always smaller than the exact value of the integral.

Which technique is useful in evaluating the integral? (a) Integration by parts (b) Partial fractions (c) Long division (d) Completing the square (e) A trig substitution (f) Other substitutions $$\int \frac{x^{2}}{1-x^{2}} d x$$

Water pipelines sometimes spring leaks, and water escapes until the leak is repaired. (a) At \(t\) days after a leak is detected, water leaks from a pipeline at an estimated rate of \(r(t)=1-\frac{1}{\sqrt{t^{2}+1}}\) thousands of gallons per day. By finding a derivative or looking at a graph, explain why this rate could represent a new leak. In the long run, if the leak is not fixed, what happens to \(r(t) ?\) (b) What is the shape of the graph of \(V(t),\) the total volume of water that has escaped by time \(t ?\) (c) Find a formula for \(V(t)\)

Calculate the integrals by partial fractions and then by using the indicated substitution. Show that the results you get are the same.$$\int \frac{2 x}{x^{2}-1} d x ; \text { substitution } w=x^{2}-1$$

Decide whether the statements are true or false. Give an explanation for your answer. \(\int 1 /\left(x^{2}+4 x-5\right) d x\) involves an arctangent.
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