Q.Integrate the following function: $x \sqrt{x+2}$

Concept: U Substitution — we substitute the inner expression of the square root to simplify the integral.

Let $u = x + 2$, so $x = u - 2$ and $dx = du$. The integral becomes:

Integrate term by term:

$\frac{u^{5/2}}{5/2} - 2 \cdot \frac{u^{3/2}}{3/2} + C = \frac{2}{5} u^{5/2} - \frac{4}{3} u^{3/2} + C$ …

Why U-Substitution?

When you see a function like $x\sqrt{x+2}$, the trouble is that $x$ and $\sqrt{x+2}$ are tangled together — one is linear in $x$, the other involves $x+2$ under a square root. You can't directly integrate a product like this using basic rules.

The trick is to untangle them. If we set $u = x+2$, then the square root becomes $\sqrt{u}$, which is clean. And $x$ becomes $u-2$, which is still just a polynomial in $u$. The whole integrand becomes a sum of powers of $u$, which we can integrate term by term.

This is the heart of substitution: you choose a new variable so that the messy part becomes simple, and the rest of the integrand (including $dx$) transforms accordingly.

Step-by-step solution

1. Choose the substitution

Let $u = x + 2$. Then $x = u - 2$, and $dx = du$.

2. Rewrite the integrand

The original integral is:

Replace $x$ with $u-2$ and $\sqrt{x+2}$ with $\sqrt{u}$:

3. Simplify the integrand algebraically

Write $\sqrt{u}$ as $u^{1/2}$:

Now it's just a sum of power functions — straightforward to integrate.

4. Integrate term by term

Use the power rule $\int u^n \, du = \frac{u^{n+1}}{n+1} + C$ for each term:

• For $u^{3/2}$: $n = \frac{3}{2}$, so $n+1 = \frac{5}{2}$. The integral is $\frac{u^{5/2}}{5/2} = \frac{2}{5} u^{5/2}$. …

Method: Substitute a linear root and re-express the leftover $x$

Use this when a lone $x$ multiplies a root of a linear expression, e.g. $x\sqrt{x+2}$: substitute the bracket and rewrite the stray $x$ in terms of $u$.

Steps

Step 1: Substitute the inside of the root.

Let $u=x+2$, so $du=dx$ and — crucially — $x=u-2$.

Step 2: Rewrite the whole integrand in $u$.

$x\sqrt{x+2}\,dx=(u-2)\,u^{1/2}\,du=(u^{3/2}-2u^{1/2})\,du.$ …

Common Mistakes

Mistake 1: Forgetting to rewrite the stray $x$ in terms of $u$.

Why it's wrong: after $u=x+2$, the leftover $x$ must become $u-2$, or the integral still contains both variables. Correct approach: use $x=u-2$ so the integrand is entirely in $u$.

Mistake 2: Treating $x\sqrt{x+2}$ as $x\cdot(x+2)^{1/2}$ and power-ruling directly.

Why it's wrong: the $x$ is not a constant, so you cannot just power-rule the root. Correct approach: substitute $u=x+2$ and expand $(u-2)u^{1/2}$. …