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Chapter 6: Problem 21

Find the integral. (Note: Solve by the simplest method-not all require integration by parts.) $$ \int x^{3} \sin x d x $$

Short Answer

Expert verified
The integral of \(\theta \sec \theta \tan \theta d \theta\) is \(\frac{1}{2}\sec^2 \theta\).

Step by step solution

01

Identify the integral rule

Recognize that the integral of \(\sec \theta \tan \theta\) with respect to \(\theta\) is \(\sec \theta\). Now, the integral given can be rewritten as \( \int \theta * (\sec \theta \tan \theta) d \theta \). This form suggests us to use the method of integration by substitution.
02

Perform substitution

Let's substitute \(u = \sec \theta\). Then, the derivative \(du\) is equal to \(\sec \theta \tan \theta d \theta\). With the substitution, our integral now becomes \(\int \theta du\).
03

Integrate using power rule

The integral \(\int \theta du\) now is a simple power rule evaluation. The power rule states that the integral of \(u^n\) is \(\frac{1}{n+1}u^{n+1}\). Therefore, the integral becomes \(\frac{1}{2}u^2\).
04

Substitute original value of \(u\)

Substitute using \(u = \sec \theta\) back into the integral to get the final answer. This results in \(\frac{1}{2}\sec^2 \theta\).

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Most popular questions from this chapter

Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by \(F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t\) if the improper integral exists. Laplace Transforms are used to solve differential equations. Find the Laplace Transform of the function. $$ f(t)=\sinh a t $$

Explain why \(\int_{-1}^{1} \frac{1}{x^{3}} d x \neq 0\)

Use a computer algebra system to find the integral. Graph the antiderivatives for two different values of the constant of integration. $$ \int \sec ^{5} \pi x \tan \pi x d x $$

Use a graphing utility to graph \(f(x)=\frac{x^{k}-1}{k}\) for \(k=1,0.1\), and 0.01 . Then evaluate the limit \(\lim _{k \rightarrow 0^{+}} \frac{x^{k}-1}{k}\).

Evaluate the definite integral. $$ \int_{0}^{\pi / 2} \frac{\cos t}{1+\sin t} d t $$
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