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Chapter 4: Problem 48

Find the indefinite integral. $$ \int x \sin x^{2} d x $$

Short Answer

Expert verified
The indefinite integral \(\int x \sin x^{2} d x = -\cos(x^{2}) / 2 + C\).

Step by step solution

01

Identify the new variable for substitution

Identify the part in the integral that will be replaced. A good choice would be \(u = x^{2}\) as \(x dx\) is another term in the integral, which will be covered by \(du\), when we differentiate \(u\). So, let \(u = x^{2}\).
02

Differentiate the substituted variable

Differentiate the substitution in respect to \(x\). So, \(du/dx = 2x\). Rearranging to replace \(dx\), we find \(dx = du/(2x)\). Here we replace \(dx\) from our integral \(\int x \sin x^{2} d x\) with \(du/(2x)\). This simplifies the integral to \(\int \sin u \cdot du/2\).
03

Perform the integration

Now, we compute the integral of the simplified function \(\int \sin u \cdot du/2\). Using basic integral identities, we know the integral of \(\sin u\) is \(-\cos u + C\), where \(C\) is the constant of integration. This step's result is \(-\cos u / 2 + C\).
04

Reverse the substitution

Finally, we reverse our substitution. Replace \(u\) back with \(x^{2}\) in the expression \(-\cos u / 2 + C\). After substitution, our final integral result is \(-\cos(x^{2}) / 2 + C\). This is the result of the indefinite integral \(\int x \sin x^{2} d x\).

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