Question

State whether you would use integration by parts to evaluate the integral. If so, identify what you would use for \(u\) and \(d v\). Explain your reasoning. $$ \int \frac{x}{\sqrt{x^{2}+1}} d x $$

Short answer

Yes, the integral \(\int \frac{x}{\sqrt{x^{2}+1}} dx\) should be solved using the integration by parts method. Set \(u = x\) and \(dv = 1 / \sqrt{x^{2}+1} dx\). The resulting integral, after applying the integration by parts formula, is \(x * \sinh^{-1}(x) - \int \sinh^{-1}(x) dx\).

Step-by-step solution

Identify the Parts for Integration

We treat the integral \(\int \frac{x}{\sqrt{x^{2}+1}} dx\) as \(\int x(1/ \sqrt{x^{2}+1}) dx\). So, we set \(u = x\) and \(dv = 1 / \sqrt{x^{2}+1} dx\).

Differentiation and Integration

Next, we differentiate \(u\) and integrate \(fv\) to get \(du = dx\) and \(v = \sinh^{-1}(x)\) where \(\sinh^{-1}(x)\) represents the inverse hyperbolic sine function.

Apply Integration by Parts

Now apply the integration by parts formula: \(\int u dv = uv - \int v du\). Substituting our values, we get: \(u v = x * \sinh^{-1}(x) - \int \sinh^{-1}(x) dx\).