Q.Integrate the following function: $\int (4 e^{3x} + 1) dx$

The key idea is that the integral of $e^{ax}$ is $\frac{1}{a} e^{ax}$, plus the constant of integration.

Step 1: Split the integral using linearity:

Step 2: Integrate each term. For $e^{3x}$, we have $\int e^{3x} \, dx = \frac{1}{3} e^{3x}$. For the constant, $\int 1 \, dx = x$. …

The key here is to recognise that integration is a linear operation. That means you can break a sum into separate pieces and pull constant factors out in front. This is the first thing to check whenever you see a sum inside an integral — don't try to integrate the whole expression at once; split it.

The two pieces we have are $4 e^{3x}$ and $1$. Each requires a different basic rule, but both are straightforward.

The exponential rule: For $\int e^{ax} dx$, the derivative of $e^{ax}$ is $a e^{ax}$, so to reverse that, you divide by $a$. Hence $\int e^{ax} dx = \frac{1}{a} e^{ax} + C$. This works for any nonzero constant $a$.

The constant rule: $\int 1 \, dx = x + C$. This is just the antiderivative of the constant function 1.

Now let's apply these step by step.

1. Split the integral.

2. Factor out the constant 4 from the first integral.

3. Apply the exponential rule to $\int e^{3x} dx$. Here $a = 3$, so:

Multiplying by the 4 we factored out:

(The constant $C_1$ gets absorbed into the overall constant later.)

4. Integrate the constant term.

5. Combine the results. $\frac{4}{3} e^{3x} + x + (C_1 + C_2)$ …

Method: Term-by-term integration using linearity

Split any sum under an integral and integrate each part with its own standard rule.

Steps

Step 1: Use linearity.

$\int[f+g]\,dx=\int f\,dx+\int g\,dx$, and pull constant multipliers outside.

Step 2: Apply each standard rule. …

Common Mistakes

Mistake 1: Writing $\int e^{3x}\,dx=\frac{1}{3x}e^{3x}$.

Why it's wrong: the exponent's coefficient is a constant $3$, so you divide by $3$, not by $x$. Correct approach: $\int e^{3x}\,dx=\frac13e^{3x}$, giving $\frac43e^{3x}$. …