Problem 44 - Essential Calculus

Question

State the method of integration you would use to perform each integration. Explain why you chose that method. Do not integrate. (a) \(\int x \sqrt{x^{2}+1} d x\) (b) \(\int x^{2} \sqrt{x^{2}-1} d x\)

Step-by-Step Solution

Answer

For integral (a), the u-substitution method would be used. For integral (b), one would use a trigonometric substitution method, specifically, secant substitution.

Step 1: Choosing the method for integral (a)

Looking at the function \(x \sqrt{x^{2}+1}\), it's observed that this could likely be a case for a simple u-substitution method. One notes that the derivative of \(x^2+1\) is \(2x\), which is also present in the integral. If \(u\) is set to be \(x^2 + 1\), then \(du = 2x dx\). This pairing of variable and its derivative makes u-substitution a suitable choice for this integral.

Step 2: Choosing the method for integral (b)

The function \(x^2 \sqrt{x^{2}-1}\) looks a bit more involved. Here, the function inside the square root, \(x^2 - 1\), has a derivative of \(2x\), which is part of the function \(x^2 \sqrt{x^{2}-1}\). However, if a simple substitution set \(u = x^2 - 1\), the remaining \(x\) would be problematic. So this integral seems to be suited for the trigonometric substitution method. In this case, since the expression under the square root is of the form \(x^2 - a^2\), this hints towards a secant substitution. Specifically, one can set: \(x= \sec(\theta)\).