Problem 42 - Essential Calculus

Question

Find or evaluate the integral using substitution first, then using integration by parts. $$ \int_{0}^{2} e^{\sqrt{2 x}} d x $$

Step-by-Step Solution

Answer

The result of the integral \(\int_{0}^{2} e^{\sqrt{2 x}} d x\) is \(2e^2-\frac{1}{2}(e^{2\sqrt{2}}-1)\)

Step 1 (Substitution)

Start the problem by setting \(u = \sqrt{2x}\). Direct calculation then gives \(du = \frac{dx}{\sqrt{2x}} = \sqrt{2} \frac{dx}{2 \sqrt{x}}\). Cross multiply to isolate \(dx\), getting \(dx = \frac{du}{\sqrt{2}} \sqrt{x}\). Substitute \(u\) back into the expression for \(\sqrt{x}\), giving \(\sqrt{x} = \frac{u}{\sqrt{2}}\). So this gives \(dx = \frac{1}{2} du\). Replace \(dx\) in the original integral and also replace the limits of integration according to \(u\). The new limits will be \(u(0)=0\) and \(u(2)=2\).

Step 2 (Integration by Substitution)

After substitution, the integral becomes: \(\int_0^{2\sqrt{2}} e^u \frac{1}{2} du\). Evaluate the integral as usual getting \(\frac{1}{2}\left[e^u\right]_0^{2\sqrt{2}} = \frac{1}{2}(e^{2\sqrt{2}}-e^0) = \frac{1}{2}(e^{2\sqrt{2}}-1)\).

Step 3 (Integration by Parts)

Now, solve the original integral with the method of parts. First, identify \(u\) and \(dv\) for the integration. Here, we can take \(u = e^{\sqrt{2x}}\) and \(dv = dx\). Now, compute \(du\) and \(v\). Computing \(du\) we get \(du = e^{\sqrt{2x}} \frac{1}{2\sqrt{x}} dx\). And \(v\) will be itself \(x\) as the integral of \(1\) with respect to \(x\) is \(x\).

Step 4 (Apply Integration By Parts Formula)

Applying the formula for integration by parts which is \(\int u dv = uv - \int v du\), our integral becomes: \(\int_{0}^{2} e^{\sqrt{2x}} dx = [xe^{\sqrt{2x}}]_{0}^{2} - \int_{0}^{2} x e^{\sqrt{2x}} \frac{1}{2\sqrt{x}} dx\). Evaluating the first term with the upper and lower limit, we get \(2e^{\sqrt{4}} - 0 = 2e^2\), simplify further to solve the integral.

Step 5 (Solve Remaining Integral Using Substitution)

The remaining integral which needs to be solved is \(- \int_{0}^{2} \frac{x}{2\sqrt{x}} e^{\sqrt{2x}} dx\). Here, again to solve this integral, we can use substitution method form step 1 and 2. After applying the same steps, the integral becomes \(-\frac{1}{2} [e^u]_0^{2\sqrt{2}} = -\frac{1}{2}(e^{2\sqrt{2}}-e^0) =-\frac{1}{2}(e^{2\sqrt{2}}-1)\).

Step 6 (Combine Results)

Add up the result of step 4 and step 5 to get the final result. So, the final result is \(2e^2-\frac{1}{2}(e^{2\sqrt{2}}-1)\)