Problem 36 - Essential Calculus

Question

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

Step-by-Step Solution

Answer

The indefinite integral of \(x \sin(x^{2}) dx\) is \(-\frac12 \cos(x^{2}) + C\).

Step 1: Identifying the Function to Substitute

Looking at the integral, it is seen that \(x^2\) is the function of \(x\) inside the sine function. By setting \(u = x^{2}\), we allow for substitution.

Step 2: Compute Derivative of Substitution Variable

Take the derivative of \(u\) with respect to \(x\). We find that \(du/dx = 2x\). To clear the derivative, we have to solve for \(dx\), which yields \(dx = du/(2x)\).

Step 3: Substitute Identified Function and Its Derivative into the Integral

Replace \(x^{2}\) with \(u\) and \(dx\) with \(du/(2x)\) in the integral. After performing these substitutions, the integral becomes: \(\int \sin(u) du/2\).

Step 4: Evaluate the Integral

The integral of \(\sin(u)\) is \(-\cos(u)\). Therefore, the result of the integral is \(-\cos(u) / 2 + C\), where \(C\) is the constant of integration.

Step 5: Use the Original Variable

Replace \(u\) with \(x^2\). This final substitution gives us \(-\frac12 \cos(x^{2}) + C\).