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Chapter 7: Problem 20

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

Short Answer

Expert verified
The integral is \(\frac{x^5}{5} + 2x^3 + 9x + C\). Verification by differentiation confirms correctness.

Step by step solution

01

Apply Binomial Expansion

First, expand the expression within the integral: \( (x^2+3)^2 \). Use the binomial formula: \((a + b)^2 = a^2 + 2ab + b^2\). Here, \(a = x^2\) and \(b = 3\), so we have \((x^2)^2 + 2 \cdot x^2 \cdot 3 + 3^2 = x^4 + 6x^2 + 9\).
02

Rewrite the Integral

Substitute the expanded expression back into the integral: \[\int (x^2+3)^2 \, dx = \int (x^4 + 6x^2 + 9) \, dx\].
03

Integrate Term by Term

Integrate each term separately: \[\int x^4 \, dx = \frac{x^5}{5}\], \[\int 6x^2 \, dx = 6 \cdot \frac{x^3}{3} = 2x^3\], and \[\int 9 \, dx = 9x\]. So, the integral becomes \[\frac{x^5}{5} + 2x^3 + 9x + C\], where \(C\) is the constant of integration.
04

Differentiate the Result to Check

Differentiate the solution \(\frac{x^5}{5} + 2x^3 + 9x + C\) with respect to \(x\): \(\frac{d}{dx}[\frac{x^5}{5} + 2x^3 + 9x + C] = x^4 + 6x^2 + 9\). This matches the integrand \((x^2 + 3)^2\), confirming that the integration is correct.

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Key Concepts

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

Integration Techniques
Integration techniques are methods used to solve integrals, which are fundamental in calculus. In solving \( \int (x^2 + 3)^2 \, dx \), we use expansion, substitution, or direct integration based on the problem. Often, breaking down complex expressions by expanding them or recognizing basic integral forms is useful.

To tackle integral problems, students can
  • Recognize basic integral forms, such as polynomials and trigonometric functions.
  • Use the power rule, which states \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) for \( n eq -1 \).
  • Apply substitution if an expression is in the form of \( u \cdot du \).
Different techniques may be combined, such as integrating each term separately after expansion. This strategy simplifies the process and makes it more manageable, especially for beginners.
Binomial Expansion
The binomial expansion is a way of expressing \( (a+b)^n \) in expanded form. It's especially important in calculus for simplifying expressions before integration or differentiation.

For the expression \( (x^2 + 3)^2 \), the binomial expansion formula helps us to avoid directly integrating a compound expression:
  • The formula \((a + b)^2 = a^2 + 2ab + b^2\) applies where \( a = x^2 \) and \( b = 3 \).
  • Expanding yields \( x^4 + 6x^2 + 9 \), a polynomial easy to integrate.
Through binomial expansions, complex algebraic expressions become simple polynomials, making them straightforward to work with in calculus.This method is powerful not just for integration, but also for simplifying problems in algebra and other areas of mathematics.
Mathematical Differentiation
Differentiation is the process of finding the derivative of a function, and it plays a critical role in calculus.By differentiating an integrated function, we can check the correctness of the integration process.Using derivative rules helps confirm solutions.

In this exercise, after integrating, we differentiate the result to verify:
  • Each term's derivative is found using the power rule, such as \( \frac{d}{dx} \left( \frac{x^5}{5} \right) = x^4 \).
  • Calculate derivatives of constants, recognizing \( \frac{d}{dx}[C] = 0 \).
  • The calculated derivative \( x^4 + 6x^2 + 9 \) matches our original expression \((x^2 + 3)^2\).
By practicing differentiation, one gains a powerful check for calculations, bridging understanding between integration and differentiation techniques.

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Most popular questions from this chapter

(a) Find \(\int \sin \theta \cos \theta d \theta\) (b) You probably solved part (a) by making the substitution \(w=\sin \theta\) or \(w=\cos \theta .\) (If not, go back and do it that way.) Now find \(\int \sin \theta \cos \theta d \theta\) by making the other substitution. (c) There is yet another way of finding this integral which involves the trigonometric identities $$ \begin{aligned}\sin (2 \theta) &=2 \sin \theta \cos \theta \\\\\cos (2 \theta) &=\cos ^{2} \theta-\sin ^{2} \theta \end{aligned}$$ Find \(\int \sin \theta \cos \theta d \theta\) using one of these identities and then the substitution \(w=2 \theta\) (d) You should now have three different expressions for the indetinite integral \(\int \sin \theta \cos \theta d \theta .\) Are they really different? Are they all correct? Explain.

Explain why the two antiderivatives are really, despite their apparent dissimilarity, different expressions of the same problem. You do not need to evaluate the integrals. $$\int \frac{\ln x}{x} d x \text { and } \, \int x \, d x$$

Solve without using integral tables. Calculate the integral \(\int \frac{1}{x^{2}-a} d x\) for (a) \(\quad a>0\) (b) \(a=0\) (c) \(a<0\)

Explain what is wrong with the statement. $$f(f(x))^{2} d x=(f(x))^{3} / 3+C$$.

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 $$\int \frac{x^{2}}{\sqrt{1-x^{2}}} d x$$
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