Problem 36 - Essential Calculus

Question

Determine which of the integrals can be found using the basic integration formulas you have studied so far in the text. (a) \(\int e^{x^{2}} d x\) (b) \(\int x e^{x^{2}} d x\) (c) \(\int \frac{1}{x^{2}} e^{1 / x} d x\)

Step-by-Step Solution

Answer

Integral (a) and Integral (c) can't be solved using the basic integration formulas, while Integral (b) can be solved with the answer as \( \frac{1}{2} e^{x^2} \).

Step 1: Evaluate integral (a)

Integral (a) is \(\int e^{x^{2}} d x\). By observing, it is immediately apparent that there is no elementary function whose derivative is \(e^{x^2}\). This means that it cannot be evaluated using the basic integration formulas that have been studied so far. Therefore, \(e^{x^{2}}\) is non-integrable in terms of elementary functions.

Step 2: Evaluate integral (b)

Integral (b) is \(\int x e^{x^{2}} d x\). In this case, a substitution can be made, let \(u = x^2\), then \(\frac{du}{dx} = 2x\) or \(dx = \frac{du}{2x}\). The integral then becomes, \(\int e^u du\), which can be solved using the basic integration formulas to get \(e^u\), the original term can be then substituted back to get the final answer as \( \frac{1}{2} e^{x^2} \). Therefore, \(x e^{x^2}\) is integrable using basic integration formulas.

Step 3: Evaluate integral (c)

Integral (c) is \(\int \frac{1}{x^{2}} e^{1 / x} d x\). The choice of substitution is not immediately clear and none of the basic integration formulas fit this equation. Thus, this integral cannot be solved using the basic formulas that have been so far studied.