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

Integrate the following function: (4x+2) √(x²+x+1)

Exercise 7.2 · Q9

(4x+2) \sqrt{x^2+x+1}

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Concept understanding — U Substitution

U Substitution: The Reverse Chain Rule

The chain rule differentiates composite functions: the derivative of $\sin(x^2)$ is $\cos(x^2) \cdot 2x$ — differentiate the outer function, then multiply by the derivative of the inside. Integration asks the reverse: given $\cos(x^2) \cdot 2x$ , find the original function. That's what u substitution does — it reverses the chain rule.

The Core Intuition

When an integral looks like "a function times the derivative of its inside," substitute the inside with $u$ and the derivative of the inside with $du$ . Consider:

\int 2x \cos(x^2) \, dx

Here $2x$ is the derivative of $x^2$ , and $x^2$ is the inside of $\cos(x^2)$ . Let $u = x^2$ , so $du = 2x \, dx$ :

\int \cos(u) \, du = \sin(u) + C = \sin(x^2) + C

Check: the derivative of $\sin(x^2)$ is $\cos(x^2) \cdot 2x$ .

The Precise Statement

\int f(g(x)) \cdot g'(x) \, dx = \int f(u) \, du \quad \text{where } u = g(x), \, du = g'(x) \, dx

Valid provided $g$ is differentiable and the resulting integral in $u$ is simpler.

The Step-by-Step Method

Identify a function $g(x)$ whose derivative $g'(x)$ also appears (possibly up to a constant factor).
Set $u = g(x)$ , compute $du = g'(x) \, dx$ .
Rewrite the entire integral in $u$ and $du$ — every $x$ and $dx$ must be replaced.
Integrate with respect to $u$ .
Substitute back $u = g(x)$ .

Watch out

You cannot mix variables. If any $x$ remains after substitution, you chose the wrong $u$ (or must solve for $x$ in terms of $u$ — rare).

A Second Example (with a constant factor)

Evaluate $\int x \sqrt{x^2 + 1} \, dx$ . Let $u = x^2 + 1$ , so $x \, dx = \frac{1}{2} du$ :

\int \sqrt{u} \cdot \tfrac{1}{2} du = \tfrac{1}{2} \cdot \tfrac{2}{3} u^{3/2} + C = \tfrac{1}{3} (x^2 + 1)^{3/2} + C

When Does It Work?

When the integrand is something times the derivative of something inside. Common patterns:

$x \cdot f(x^2)$ — derivative of $x^2$ is $2x$ , so $u = x^2$
$e^{g(x)} \cdot g'(x)$ — derivative of $g(x)$ appears
$\frac{g'(x)}{g(x)}$ — leads to $\log|g(x)|$ …

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The integral is solved by recognising that $4x+2$ is exactly $2$ times the derivative of $x^2+x+1$ , making a direct $u$ -substitution possible. The result is $\frac{4}{3}(x^2+x+1)^{3/2} + C$ .

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

Choose the substitution.

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

Then $du = (2x+1)\,dx$ .
Rewrite the integrand in terms of $u$ .

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

4x+2 = 2(2x+1).

So the integral becomes

\int 2(2x+1) \sqrt{x^2+x+1}\,dx.

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

\int 2 \sqrt{u}\,du = 2 \int u^{1/2}\,du.

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

2 \cdot \frac{u^{3/2}}{3/2} + C = 2 \cdot \frac{2}{3} u^{3/2} + C = \frac{4}{3} u^{3/2} + C.

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

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

Concept understanding — U Substitution

U Substitution: The Reverse Chain Rule

The Core Intuition

The Precise Statement

The Step-by-Step Method

A Second Example (with a constant factor)

When Does It Work?

Why substitution works here

Step-by-step solution

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

Steps

Common Mistakes

Q.Integrate the following function: (4x+2)x2+x+1(4x+2) \sqrt{x^2+x+1}(4x+2)x2+x+1​

Why substitution works here

Step-by-step solution

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

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