Question

Evaluate the following integrals: $$ \int(x+2) \sqrt{2 x^{2}+2 x+1} d x $$

Short answer

Question: Evaluate the integral $$\int (x+2) \sqrt{2x^2 + 2x + 1} \,dx.$$ Answer: $$\frac{1}{3}(2x^2 + 2x + 1)^{\frac{3}{2}} + C$$

Step-by-step solution

Identify the integrand and the suitable substitution

The given integral is: $$ \int (x+2) \sqrt{2x^2 + 2x + 1} \,dx $$ We observe that the expression inside the square root is a quadratic in the form of \((ax^2 + bx + c)\). A good approach for such integrals is to substitute the expression inside the square root with a variable squared, so we can use the substitution. Let \(u^2 = 2x^2 + 2x + 1\).

Apply the substitution

We need to find the derivative of this substitution with respect to \(x\), which is \(du/dx\). Differentiating both sides with respect to \(x\), we get: $$ 2u\frac{du}{dx} = 4x + 2 $$ Now, solve for \(\frac{du}{dx}\): $$ \frac{du}{dx} = \frac{4x+2}{2u} $$ We can rewrite \(dx\) in terms of \(du\): $$ dx = \frac{2u}{4x+2} \, du $$ Now, we substitute \(u^2\) and \(dx\) in the original integral: $$ \int (x+2) \sqrt{u^2} \cdot \frac{2u}{4x+2} \, du $$

Evaluate the resulting integral

After the substitution, we can simplify the integral as follows: $$ \int u^2 \, du $$ Now, let's evaluate the integral: $$ \frac{1}{3}u^3 + C $$

Substitute back to the original variable

Recall that \(u^2 = 2x^2 + 2x + 1\). Now, we can substitute back to find the integral in terms of \(x\): $$ \frac{1}{3}(2x^2 + 2x + 1)^{\frac{3}{2}} + C $$

Present the final result

The evaluated integral is: $$ \int (x+2) \sqrt{2x^2 + 2x + 1} \,dx = \frac{1}{3}(2x^2 + 2x + 1)^{\frac{3}{2}} + C $$