Q.$\int(2x - 3\cos x + e^x)\,dx =$ __________.

The Intuition: "Differentiation distributes, so integration should too"

Suppose your speed has two parts: you speed up from excitement (part A) and slow from tiredness (part B). Your total speed is the sum. Your total distance — the antiderivative of speed — is then the distance from part A plus the distance from part B. That's the core idea: the antiderivative of a sum is the sum of the antiderivatives.

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The Precise Statement

The indefinite integral of a sum of two functions equals the sum of their individual antiderivatives. This holds for any $f$ and $g$ that have antiderivatives. The same rule applies to subtraction:

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Why It's True (A Quick Proof)

Let $F'(x) = f(x)$ and $G'(x) = g(x)$. Consider $H(x) = F(x) + G(x)$:

So $H(x)$ is an antiderivative of $f(x)+g(x)$ — exactly the statement.

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A Concrete Example

Find $\int (x^2 + \cos x) \, dx$.

Step 1: Apply the sum rule: $\int x^2 \, dx + \int \cos x \, dx$

Step 2: Each antiderivative: $\int x^2 \, dx = \frac{x^3}{3}$, $\int \cos x \, dx = \sin x$

Step 3: Combine:

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Why This Matters for Exams

The antiderivative of a sum is the first tool for any integral that isn't a single standard form. It lets you break $\int (3x^2 + 2x + 1)\,dx$ into three easy integrals, or split $\int (\sin x + e^x)\,dx$ into known results.

Common mistake: trying to apply it to products or quotients. It does not work there: …