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Chapter 7: Problem 2

Before integrating, how would you rewrite the integrand of \(\int\left(x^{4}+2\right)^{2} d x ?\)

Short Answer

Expert verified
Question: Rewrite the integrand of the integral \(\int\left(x^{4}+2\right)^{2} dx\) and simplify the expression. Answer: The rewritten integrand is \(x^8 + 4x^4 + 4\), so the integral becomes \(\int\left(x^{8}+4x^{4}+4\right) dx\).

Step by step solution

01

Expand the squared term

We will start by expanding the integrand \((x^4 + 2)^2\) using the binomial formula: \((a+b)^2 = a^2 + 2ab + b^2\). Here, \(a = x^4\) and \(b = 2\). \((x^4 + 2)^2 = (x^4)^2 + 2 \cdot (x^4) \cdot 2 + 2^2\)
02

Simplify the expression

Now, we can simplify the expanded expression: \((x^4)^2 = x^8\) \(2 \cdot (x^4) \cdot 2 = 4x^4\) \(2^2 = 4\) So, the simplified expression is: \(x^8 + 4x^4 + 4\) The rewritten integrand is thus \(x^8 + 4x^4 + 4\). Therefore, the integral becomes: \(\int\left(x^{8}+4x^{4}+4\right) dx\)

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Most popular questions from this chapter

Graph the function \(f(x)=\frac{1}{x \sqrt{x^{2}-36}}\) on its domain. Then find the area of the region \(R_{1}\) bounded by the curve and the \(x\) -axis on \([-12,-12 / \sqrt{3}]\) and the area of the region \(R_{2}\) bounded by the curve and the \(x\) -axis on \([12 / \sqrt{3}, 12] .\) Be sure your results are consistent with the graph.

Use numerical methods or a calculator to approximate the following integrals as closely as possible. $$\int_{0}^{\infty} \ln \left(\frac{e^{x}+1}{e^{x}-1}\right) d x=\frac{\pi^{2}}{4}$$

For what values of \(p\) does the integral \(\int_{2}^{\infty} \frac{d x}{x \ln ^{p} x}\) exist and what is its value (in terms of \(p\) )?

Suppose that a function \(f\) has derivatives of all orders near \(x=0 .\) By the Fundamental Theorem of Calculus, \(f(x)-f(0)=\int_{0}^{x} f^{\prime}(t) d t\) a. Evaluate the integral using integration by parts to show that $$f(x)=f(0)+x f^{\prime}(0)+\int_{0}^{x} f^{\prime \prime}(t)(x-t) d t.$$ b. Show (by observing a pattern or using induction) that integrating by parts \(n\) times gives $$\begin{aligned} f(x)=& f(0)+x f^{\prime}(0)+\frac{1}{2 !} x^{2} f^{\prime \prime}(0)+\cdots+\frac{1}{n !} x^{n} f^{(n)}(0) \\ &+\frac{1}{n !} \int_{0}^{x} f^{(n+1)}(t)(x-t)^{n} d t+\cdots \end{aligned}$$ This expression is called the Taylor series for \(f\) at \(x=0\).

Use symmetry to evaluate the following integrals. a. \(\int_{-\infty}^{\infty} e^{|x|} d x \quad\) b. \(\int_{-\infty}^{\infty} \frac{x^{3}}{1+x^{8}} d x\)
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