Q.Integrate the following function: $\int (1 - x) \sqrt{x} dx$

The key idea is to rewrite the integrand as a sum of power functions, then apply the Power Rule for integration term by term.

First, rewrite $\sqrt{x}$ as $x^{1/2}$ and expand:

Now integrate each term using $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$:

$\int x^{3/2} \, dx = \frac{x^{5/2}}{5/2} = \frac{2}{5} x^{5/2}.$ …

Why This Approach Works

When you see a product like $(1 - x)\sqrt{x}$, your first instinct might be to look for a substitution. But here, the expression is already a sum of simple power functions once you multiply it out. The square root $\sqrt{x}$ is just $x^{1/2}$, so the whole integrand becomes a combination of $x^{1/2}$ and $x^{3/2}$ terms.

The Power Rule for integration says: for any real number $n \neq -1$,

This is the direct reverse of the differentiation rule $\frac{d}{dx}x^n = n x^{n-1}$. Since both exponents here ($1/2$ and $3/2$) are not $-1$, we can integrate each term separately.

Step-by-Step Solution

1. Expand the integrand.

Multiply $(1 - x)$ by $\sqrt{x}$:

Since $x \cdot x^{1/2} = x^{1 + 1/2} = x^{3/2}$, we have:

2. Write the integral as a sum of two power terms.

3. Apply the Power Rule to each term. …

Method: Expand-then-Power-Rule for products and roots

Use this whenever the integrand is a product or a root of $x$ (like $(1-x)\sqrt{x}$) that has no single standard formula — the trick is to turn it into a plain sum of powers first.

Steps

Step 1: Rewrite every root as a fractional power.

Replace $\sqrt{x}$ by $x^{1/2}$, $\dfrac{1}{\sqrt{x}}$ by $x^{-1/2}$, and so on, so the whole integrand is written using $x^n$.

Step 2: Multiply out into a sum of pure power terms.

Distribute the product; use the exponent law $x^a\cdot x^b = x^{a+b}$ so each piece becomes a single $x^n$. …

Common Mistakes

Mistake 1: Treating the integral of a product as a product of integrals.

Why it's wrong: $\int (1-x)\sqrt{x}\,dx \neq \big(\int(1-x)\,dx\big)\big(\int\sqrt{x}\,dx\big)$ — there is no product rule for integration. Correct approach: expand $(1-x)\sqrt{x}=x^{1/2}-x^{3/2}$ first, then integrate each power.

Mistake 2: Dividing by the old exponent instead of the new one. …