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Chapter 6: Problem 16

Use integration tables to find the integral. $$ \int \sqrt{x} \arctan x^{3 / 2} d x $$

Short Answer

Expert verified
The integral \( \int \sqrt{x} \arctan x^{3 / 2} dx = \arctan x^{3 / 2} * 2/3 x^{3 / 2} - 2/3 \arctan x^{3 / 2} + c\)

Step by step solution

01

Identify the parts for integration by parts

Integration by parts is a technique of integration, which is useful when the integral is a product of two functions, one of which is easily integrated. The integral \( \int \sqrt{x} \arctan x^{3 / 2} dx \) appears to be a product of \( \arctan x^{3 / 2} \) and \( \sqrt{x} \). The method requires us to identify two parts in the integral: a part that can be easily integrated which we will call 'dv', and a part when differentiated becomes simpler which we will call 'u'. Here, it can be seen that \(dv = \sqrt{x}dx\) and \(u = \arctan x^{3 / 2} \).
02

Calculate du and V

We will now calculate du which is the derivative of u and V which is the integral of dv. From \(u = \arctan x^{3 / 2}\), we differentiate u to get \(du = 1/(1+(x^{3 / 2})^2) * 3 / 2 \sqrt{x} dx\). Taking the integral of dv = \( \sqrt{x}dx \) will yield \(V = 2/3 x^{3 / 2}\).
03

Apply the integration by parts formula

Now we can apply the integration by parts formula: \( \int u dv = uV - \int V du\). This will give us \( \int \sqrt{x} \arctan x^{3 / 2} dx = \arctan x^{3 / 2} * 2/3 x^{3 / 2} - \int 2/3 x^{3 / 2} * 1/(1+(x^{3 / 2})^2)* 3 / 2 \sqrt{x} dx\).
04

Simplify the integral

The integral can be simplified to \( \int x^{3 / 2}/(1+(x^{3 / 2})^2) dx\). Now apply the substitution \( t = x^{3 / 2} \) to make \( dt = 3/2 \sqrt{x} dx \), or equivalently \( dx = 2/3 dt/\sqrt{t} \). This makes the new integral \(2/3 \int dt/(1+t^2) \). The integral of 1/(1 + t^2) is a standard integral and it is \(\arctan t\). Therefore, this term integrates to \(2/3 \arctan t + c\), where c is the constant of integration.
05

Substitute back for t

Remember that we let \(t = x^{3 / 2}\). Substituting back gives \(2/3 \arctan x^{3 / 2} + c\). This gives the final answer: \( \int \sqrt{x} \arctan x^{3 / 2} dx = \arctan x^{3 / 2} * 2/3 x^{3 / 2} - 2/3 \arctan x^{3 / 2} + c\)

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Most popular questions from this chapter

Sketch the graph of \(g(x)=\left\\{\begin{array}{ll}e^{-1 / x^{2}}, & x \neq 0 \\ 0, & x=0\end{array}\right.\) and determine \(g^{\prime}(0)\).

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Consider the integral \(\int_{0}^{\pi / 2} \frac{4}{1+(\tan x)^{n}} d x\) where \(n\) is a positive integer. (a) Is the integral improper? Explain. (b) Use a graphing utility to graph the integrand for \(n=2,4,\) \(8,\) and \(12 .\) (c) Use the graphs to approximate the integral as \(n \rightarrow \infty\). (d) Use a computer algebra system to evaluate the integral for the values of \(n\) in part (b). Make a conjecture about the value of the integral for any positive integer \(n\). Compare your results with your answer in part (c).

(a) Let \(f^{\prime}(x)\) be continuous. Show that \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x-h)}{2 h}=f^{\prime}(x)\) (b) Explain the result of part (a) graphically.

Use a computer algebra system to evaluate the definite integral. In your own words, describe how you would integrate \(\int \sec ^{m} x \tan ^{n} x d x\) for each condition. (a) \(m\) is positive and even. (b) \(n\) is positive and odd. (c) \(n\) is positive and even, and there are no secant factors. (d) \(m\) is positive and odd, and there are no tangent factors.
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