The power rule for $\int x^n \, dx$ requires $n \neq -1$. The antiderivative of $\frac{1}{x}$ is $\ln\lvert x \rvert$, not $\frac{x^0}{0}$.

Question 1

What is example 1. Polynomial?

Accepted Answer

$\int (5 x^4 - 6 x^2 + 7) \, dx = x^5 - 2 x^3 + 7 x + c$.

Question 2

What is example 2. Rational and root?

Accepted Answer

$\int \left( \frac{1}{x^2} + \sqrt{x} \right) dx$.

Question 3

What is example 3. Exponential and reciprocal?

Accepted Answer

$\int \left( e^{-2x} + \frac{4}{x} \right) dx = -\frac{1}{2} e^{-2x} + 4 \ln\lvert x \rvert + c$.

Question 4

What is example 4. Circular with non-unit coefficient?

Accepted Answer

$\int 3 \cos(4x) \, dx = 3 \cdot \frac{1}{4} \sin(4x) + c = \frac{3}{4} \sin(4x) + c$.

Question 5

What is example 5. Initial value problem?

Accepted Answer

Given $f'(x) = 6 x - \sin(x)$ and $f(0) = 4$.

Question 6

What is forgetting the $\frac{1}{k}$ factor?

Accepted Answer

$\int e^{2x} \, dx = \frac{1}{2} e^{2x} + c$, not $e^{2x} + c$. Same for $\sin(kx)$ and $\cos(kx)$.

Question 7

What is wrong sign on $\cos$ antiderivative?

Accepted Answer

$\int \cos(x) \, dx = \sin(x) + c$ (no negative). $\int \sin(x) \, dx = -\cos(x) + c$ (negative).

Question 8

What is forgetting $c$?

Accepted Answer

An indefinite integral without the constant of integration loses marks even when the antiderivative is otherwise correct.

Question 9

What is missing the absolute value on $\ln$?

Accepted Answer

$\int \frac{1}{x} \, dx = \ln\lvert x \rvert + c$. Without context restricting $x > 0$, the absolute value is required.

Question 10

What is trying to apply the power rule to $\frac{1}{x}$?

Accepted Answer

The power rule for $\int x^n \, dx$ requires $n 
eq -1$. The antiderivative of $\frac{1}{x}$ is $\ln\lvert x vert$, not $\frac{x^0}{0}$.

Question 11

What is confusing derivative and antiderivative?

Accepted Answer

$\frac{d}{dx}[e^{2x}] = 2 e^{2x}$ (multiply by 2). $\int e^{2x} \, dx = \frac{1}{2} e^{2x} + c$ (divide by 2). The factor goes opposite directions.