# Antiderivatives and Indefinite Integrals

Mathematics I · Integral Calculus · https://tryals.app/learn/mathematics-i/antiderivatives-and-indefinite-integrals

## Undoing Derivatives

**Integration** reverses differentiation. An **antiderivative** of $f$ is a function $F$ where $F'(x) = f(x)$. Where derivatives ask for rates of change, antiderivatives recover the original function from its rate.

Antiderivatives are **not unique**. Since any constant has a zero derivative, if $F$ is an antiderivative, then $F + C$ is also one. The **indefinite integral** represents this entire family of parallel curves:

$$\int f(x)\, dx = F(x) + C$$

| Rule | Formula | Notes |
| --- | --- | --- |
| **Power Rule** | $\int x^n\, dx = \frac{x^{n+1}}{n+1} + C$ | For $n \neq -1$ |
| **Reciprocal** | $\int \frac{1}{x}\, dx = \ln|x| + C$ | Handles $n = -1$ |
| **Exponential** | $\int e^x\, dx = e^x + C$ | Unchanged |
| **Cosine** | $\int \cos x\, dx = \sin x + C$ | Reverses sine derivative |

## Rules and Pitfalls

Linearity rules carry over directly from differentiation: constants factor out, and the integral of a sum is the sum of the individual integrals.

More advanced techniques like **substitution** (reversing the chain rule) and **integration by parts** (reversing the product rule) handle complex cases later.

| Common Mistake | Why It Happens | Correct Approach |
| --- | --- | --- |
| **Omitting $+C$** | Treating the integral as a single static function | Always append $+C$ to represent the infinite vertical family |
| **Power Rule Slip** | Differentiating instead of integrating | **Raise** the exponent and divide by the new power |

## Practice questions

4 of this lesson's 10 practice questions, with answers. The full set is in the app.

### 1. Indefinite integrals always carry '+ C'. Why is dropping it a real error rather than untidiness?

A. Because the constant is needed to make the units work out
B. Because differentiation destroys constants, so the family shares one
C. Because omitting it makes the answer wrong by exactly one
D. Because C is the value of the integral at zero, which must be stated

**Answer:** B. Because differentiation destroys constants, so the family shares one

**Why:** Every vertical shift has the same slope everywhere, so differentiation loses the constant and no antiderivative can recover it. The C is the honest record of that loss, and in a differential equation it is where the initial condition goes, which is why setting it to zero silently assumes the system started at the origin.

Page: https://tryals.app/practice/mathematics-i/antiderivatives-and-indefinite-integrals/indefinite-integrals-always-carry-c-why-is-dropping-it-a-real

### 2. Select every function whose derivative is 3x^2.

A. 3x^3
B. x^3 - 2
C. x^3 + 7
D. x^3
E. 9x

**Answer:** B. x^3 - 2; C. x^3 + 7; D. x^3

**Why:** $x^{3}$, $x^{3}+7$ and $x^{3}-2$ all differentiate to $3x^{2}$, they are members of one vertical family. $3x^{3}$ gives $9x^{2}$ and 9x gives 9. This is the family that + C names.

Page: https://tryals.app/practice/mathematics-i/antiderivatives-and-indefinite-integrals/select-every-function-whose-derivative-is-3x-2

### 3. Match each integrand to its antiderivative.

**Answer:**

- 1/x → ln|x| + C
- e^x → e^x + C
- cos x → sin x + C
- x^2 → x^3/3 + C

**Why:** These four are the working vocabulary of the lesson. Note that $e^{x}$ is its own antiderivative, the only function (up to a factor) with that property, and that 1/x needs the absolute value, since ln is undefined for negative arguments.

Page: https://tryals.app/practice/mathematics-i/antiderivatives-and-indefinite-integrals/match-each-integrand-to-its-antiderivative

### 4. Order the steps for finding the one antiderivative of f that passes through a given point.

**Answer:**

1. Integrate f term by term using the power rule and the standard forms
2. Write the general antiderivative F(x) with its constant of integration C
3. Substitute the given point's coordinates into F(x)
4. Solve the resulting equation for C
5. Write out the particular antiderivative with C replaced by its value

**Why:** Integration produces a family; the point selects one member. Attempting to substitute before writing + C leaves nothing to solve for, which is why the constant must be carried from the very first line.

Page: https://tryals.app/practice/mathematics-i/antiderivatives-and-indefinite-integrals/order-the-steps-for-finding-the-one-antiderivative-of-f-that-passes