Indefinite integrals and standard forms

If a derivative tells you how fast something changes, an antiderivative recovers the original quantity from its rate of change. Given $f$, we look for $F$ with $F'(x) = f(x)$. The collection of all such $F$ is the indefinite integral, written $\int f(x)\,dx$. It is a family of functions , any two members differ only by an additive constant , because differentiation kills constants.

Definitions

An antiderivative of $f$ on an interval $I$ is a function $F$ such that $F'(x) = f(x)$ for every $x \in I$. The indefinite integral is $\int f(x)\,dx = F(x) + C,$ where $C \in \mathbb{R}$ is the constant of integration. The symbol $\int$ comes from a stretched "S" (for sum), $f(x)$ is the integrand, and $dx$ indicates the variable of integration.

Why the "$+C$"?

Suppose $F'(x) = G'(x) = f(x)$ on an interval. Then $(F - G)'(x) = 0$, so $F - G$ is constant on that interval. Hence any two antiderivatives differ by a constant. We capture all of them by writing $F(x) + C$.

The constant is more than a formality. In applied problems , e.g. recovering position from velocity , the constant encodes the initial condition. Without it, the answer is ambiguous. Always write the $+C$ in indefinite integrals.

Linearity

Integration distributes over sums and pulls out constants: $\int [af(x) + bg(x)]\,dx = a\int f(x)\,dx + b\int g(x)\,dx.$

This is the workhorse for splitting a complicated integrand into easy pieces. The identity is immediate from the corresponding property of derivatives, applied in reverse.

Note that integration does not play nicely with products or quotients. There is no simple "integral of a product" rule , you must use integration by parts. There is no quotient rule for integrals , you may need partial fractions, substitution, or other tools.

The standard integral table

Every entry below is a derivative formula read backwards. Verify each one by differentiating the right side.

A few remarks. The integral of $\sin x$ is $-\cos x$ (mind the minus sign , a common error). $\int \tan x\,dx$ may equivalently be written as $-\ln|\cos x|+C$, which is the form some prefer; both are correct. The formula for $\int \sec x\,dx$ looks mysterious , it is derived by multiplying numerator and denominator by $\sec x + \tan x$ and recognising a $u$-substitution.

Worked examples

Example 1. Compute $\int (3x^2 - 4\sin x + e^x)\,dx$.

By linearity: $\int 3x^2\,dx = x^3$, $\int -4\sin x\,dx = 4\cos x$, $\int e^x\,dx = e^x$. So the integral is $x^3 + 4\cos x + e^x + C$.

Example 2. Compute $\int \dfrac{x^2 + 1}{x}\,dx$.

Split: $\dfrac{x^2+1}{x} = x + \dfrac{1}{x}$. Integrate term by term: $\dfrac{x^2}{2} + \ln|x| + C$.

Example 3. Compute $\int \dfrac{1 - \cos 2x}{2}\,dx$.

Recognise $\dfrac{1 - \cos 2x}{2} = \sin^2 x$. But it is easier to integrate directly: $\int \dfrac{1}{2}\,dx - \int \dfrac{\cos 2x}{2}\,dx = \dfrac{x}{2} - \dfrac{\sin 2x}{4} + C$.

Example 4. Compute $\int \tan^2 x\,dx$.

Use $\tan^2 x = \sec^2 x - 1$. Then $\int \tan^2 x\,dx = \tan x - x + C$.

Example 5. Find $F(x)$ with $F'(x) = 3x^2 + 2$ and $F(1) = 5$.

Antiderivative: $F(x) = x^3 + 2x + C$. Apply the condition: $1 + 2 + C = 5$, so $C = 2$. Therefore $F(x) = x^3 + 2x + 2$.

Example 6. Compute $\int (e^x + 5)^2\,dx$.

Expand: $e^{2x} + 10e^x + 25$. Integrate: $\dfrac{e^{2x}}{2} + 10e^x + 25x + C$.

Try it yourself

1. $\int (5x^4 - 3x^2 + 7)\,dx$
2. $\int \dfrac{x^3 - 2x + 1}{x^2}\,dx$
3. $\int \cos^2 x\,dx$ (use $\cos^2 x = \tfrac{1+\cos 2x}{2}$)
4. $\int \sin^2 x\,dx$
5. $\int (\sec x + \tan x)\sec x\,dx$
6. $\int \dfrac{1 + \sin x}{\cos^2 x}\,dx$
7. $\int 3^x\,dx$
8. $\int (e^x - 1)^2 e^x\,dx$ (hint: anticipate substitution but try expanding first)
9. $\int (\sqrt{x} + 1)^2\,dx$
10. Find $f$ if $f'(x) = \cos x + \sec^2 x$ and $f(0) = 1$.
11. $\int \cot^2 x\,dx$
12. $\int \dfrac{1 - \cos x}{\sin^2 x}\,dx$ (use $1 - \cos x = 2\sin^2(x/2)$)
13. $\int x^{-3/2}\,dx$
14. Verify $\int \sec x\,dx = \ln|\sec x + \tan x| + C$ by differentiating the right side.

Pitfalls and tricks

• Forgetting $+C$: half a mark in board exams, every time.
• Sign on $\int \sin x\,dx$: the answer is $-\cos x$, not $\cos x$.
• $\int \dfrac{dx}{x}$ is $\ln|x|$, not $\ln x$ , the absolute value matters because the function is defined for $x < 0$ too.
• The power rule fails at $n = -1$ , that case is the log.
• Simplify first: many integrands look hopeless until you split a fraction, expand a square, or use a trigonometric identity. Always try algebraic simplification before reaching for a heavy technique.