Q.Integrate the following function: $(4x+2) \sqrt{x^2+x+1}$

Concept: U Substitution — the derivative of $x^2 + x + 1$ is $2x+1$, which is half of $4x+2$.

Let $u = x^2 + x + 1$. Then $du = (2x+1)\,dx$. Notice that $4x+2 = 2(2x+1)$, so:

The integral becomes:

Integrate: …

Why substitution works here

When you see a product like $(4x+2) \sqrt{x^2+x+1}$, your first instinct should be to check if one factor is the derivative of the expression inside the other. Here, the derivative of $x^2+x+1$ is $2x+1$. Notice that $4x+2 = 2(2x+1)$ — that’s exactly twice the derivative. This is the hallmark of a substitution that will simplify the integral completely, because the chain rule in reverse tells us that if we set $u = x^2+x+1$, then $du = (2x+1)\,dx$, and we have a perfect match.

Step-by-step solution

1. Choose the substitution.

   Let $u = x^2 + x + 1$.

   Then $du = (2x+1)\,dx$.

2. Rewrite the integrand in terms of $u$.

   The integrand is $(4x+2)\sqrt{x^2+x+1}$. Factor the $4x+2$:

So the integral becomes

3. Substitute $u$ and $du$. Replace $(2x+1)\,dx$ with $du$, and $\sqrt{x^2+x+1}$ with $\sqrt{u}$:

4. Integrate with respect to $u$. Using the power rule $\int u^n\,du = \frac{u^{n+1}}{n+1} + C$:

5. Substitute back for $x$. Since $u = x^2 + x + 1$, we have $\frac{4}{3} (x^2 + x + 1)^{3/2} + C.$ …

Method: Match the numerator to the derivative of the inside of the root

Use this when a factor out front is a constant multiple of the derivative of the expression under the root — here $4x+2=2(2x+1)$ and $\dfrac{d}{dx}(x^2+x+1)=2x+1$.

Steps

Step 1: Check the derivative match.

Differentiate the inside: $\dfrac{d}{dx}(x^2+x+1)=2x+1$. Since $4x+2=2(2x+1)$, the front factor is $2\times$ that derivative.

Step 2: Substitute the inside. …

Common Mistakes

Mistake 1: Not seeing that $4x+2=2(2x+1)$ is twice the derivative of the radicand.

Why it's wrong: missing this means missing the substitution entirely. Correct approach: differentiate $x^2+x+1$ to get $2x+1$, then note the front factor is $2\times$ it.

Mistake 2: Forgetting the factor $2$ from $(4x+2)\,dx=2\,du$. …