Q.Integrate the following function: $\int (2x^2 + e^x) dx$

The key idea is Power Rule Integration (for $x^n$) and the standard integral of $e^x$.

Step 1: Split the integral using linearity:

Step 2: Apply the Power Rule $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ for $n=2$:

$2 \cdot \frac{x^{3}}{3} = \frac{2}{3}x^3$ …

Why This Works

Integration is the reverse of differentiation. When you see a sum like $2x^2 + e^x$, you can break it apart because the derivative of a sum is the sum of derivatives — so the integral works the same way. Each term has its own standard rule.

The Power Rule for Integration says: to integrate $x^n$, increase the exponent by 1 and divide by the new exponent. That is,

For $e^x$, the rule is even simpler: the integral of $e^x$ is itself, $e^x + C$, because the derivative of $e^x$ is $e^x$.

The constant $C$ is crucial — it accounts for any constant that might have been lost when differentiating. Without it, your answer is incomplete.

Step-by-Step Solution

1. Separate the integral Since integration distributes over addition:

2. Integrate the first term: $2x^2$ Pull the constant 2 outside the integral:

Apply the Power Rule with $n = 2$:

So the first term becomes:

3. Integrate the second term: $e^x$ The integral of $e^x$ is simply $e^x$: $\int e^x \, dx = e^x.$ …

Method: Linearity with mixed power and exponential terms

For $\int(\text{power}+e^x)\,dx$, integrate each term with its own rule.

Steps

Step 1: Separate the terms.

$\int(2x^2+e^x)\,dx=2\int x^2\,dx+\int e^x\,dx$.

Step 2: Apply the standard rules. …

Common Mistakes

Mistake 1: Applying the power rule to $e^x$.

Why it's wrong: $\int e^x\,dx=e^x$, not $\frac{e^{x+1}}{x+1}$; the power rule is only for $x^n$ (a variable base with constant power). Correct approach: use $\int e^x\,dx=e^x$. …