Q.Integrate the following function: $\int (2x^2 - 3\sin x + 5\sqrt{x}) dx$

The key idea is to integrate term-by-term using the Power Rule for $x^n$ and the standard integral of $\sin x$.

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

Now integrate each term separately:

• $\int 2x^2\,dx = 2 \cdot \frac{x^{3}}{3} = \frac{2}{3}x^3$
• $\int -3\sin x\,dx = -3(-\cos x) = 3\cos x$
• $\int 5x^{1/2}\,dx = 5 \cdot \frac{x^{3/2}}{3/2} = 5 \cdot \frac{2}{3}x^{3/2} = \frac{10}{3}x^{3/2}$ …

The key idea is that integration is linear — you can break a sum into separate integrals and handle each piece with its own rule. This problem gives you three very different terms: a polynomial term, a trigonometric term, and a radical term. Each one uses a standard integration formula, so there’s no trick, just careful application.

1. Integrate $2x^2$

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

Here $n = 2$, so

Don’t forget the constant of integration — we’ll add it at the very end.

2. Integrate $-3\sin x$

You should know that

So

A common slip is to forget the sign change — the derivative of $\cos x$ is $-\sin x$, so the integral of $\sin x$ must be $-\cos x$.

3. Integrate $5\sqrt{x}$ …

Method: Term-by-term integration of a mixed sum

Apply this when one integral mixes different families — a polynomial term, a trig term, and a root — such as $2x^2-3\sin x+5\sqrt{x}$. Linearity lets each term use its own rule.

Steps

Step 1: Separate the sum and factor out coefficients.

Step 2: Match each term to its standard antiderivative. …

Common Mistakes

Mistake 1: Writing $\int\sin x\,dx=\cos x$ (missing the minus sign).

Why it's wrong: $\dfrac{d}{dx}\cos x=-\sin x$, so $\int\sin x\,dx=-\cos x$; hence $\int(-3\sin x)\,dx=+3\cos x$. Correct approach: watch the double sign flip.

Mistake 2: Mis-dividing when integrating $5\sqrt{x}$. …