This lesson gathers every integration rule from the series into one place: the constant and power rules, the logarithmic rule that covers the gap they leave, the three exponential cases, the properties that let you break an integral apart, and definite integrals. It ends with the one mistake that costs marks more than any other — splitting a product the way you would split a sum.

Definition Every rule in one place

Constant

\displaystyle\int k\,dx = kx + C

Power rule

\displaystyle\int x^{n}\,dx = \dfrac{x^{n+1}}{n+1} + C

n \neq -1

Logarithmic

\displaystyle\int \dfrac{1}{x}\,dx = \ln|x| + C

Exponential, base

e

\displaystyle\int e^{x}\,dx = e^{x} + C

with a coefficient

\displaystyle\int e^{kx}\,dx = \dfrac{e^{kx}}{k} + C

Exponential, base

a

\displaystyle\int a^{x}\,dx = \dfrac{a^{x}}{\ln a} + C

a > 0,\; a \neq 1

Constant multiple

\displaystyle\int k\,f(x)\,dx = k\int f(x)\,dx

Sum or difference

\displaystyle\int \big(f \pm g\big)\,dx = \int f\,dx \pm \int g\,dx

Definite integral

\displaystyle\int_{a}^{b} f(x)\,dx = F(b) - F(a)

Reversed limits

\displaystyle\int_{b}^{a} f(x)\,dx = -\int_{a}^{b} f(x)\,dx

Note Where the power rule breaks

The power rule adds one to the exponent and divides by the result. At $n = -1$ the new exponent is zero:

\displaystyle\int x^{-1}\,dx = \dfrac{x^{-1+1}}{-1+1} = \dfrac{x^{0}}{0}

Division by zero is undefined, so the rule cannot reach this one case. But $x^{-1}$ is $\dfrac{1}{x}$ , and that is exactly what the logarithmic rule integrates: $\ln|x| + C$ . The one exponent the power rule misses is the one the logarithmic rule was made for.

Note Power or exponential?

The difference is where the variable sits. In $x^{n}$ the variable is in the base and the exponent is fixed — that is a power, and the power rule applies. In $a^{x}$ the variable is in the exponent and the base is fixed — that is an exponential, and it has its own rule.

\displaystyle\int x^{2}\,dx = \dfrac{x^{3}}{3} + C

— power

\displaystyle\int 2^{x}\,dx = \dfrac{2^{x}}{\ln 2} + C

— exponential

Note Products do not split

The properties let you pull constants out and break sums and differences apart. They do not extend to multiplication:

\int f(x)\,g(x)\,dx \;\neq\; \left( \int f(x)\,dx \right)\!\left( \int g(x)\,dx \right)

If the product simplifies, simplify it first and integrate once. If it does not, that is what integration by parts is for — a later lesson.

Example The power rule, five ways

\displaystyle\int 7\,dx

— a constant integrates directly:

⟹ $7x + C$

\displaystyle\int x^{4}\,dx

— add one, divide by five:

⟹ $\dfrac{x^{5}}{5} + C$

\displaystyle\int 6x^{2}\,dx

— pull the

6

out, then

\dfrac{x^{3}}{3}

⟹ $2x^{3} + C$

\displaystyle\int \sqrt{x}\,dx = \int x^{1/2}\,dx

— new exponent

\tfrac{3}{2}

, divide by it:

⟹ $\dfrac{2}{3}x^{3/2} + C$

\displaystyle\int \dfrac{1}{x^{3}}\,dx = \int x^{-3}\,dx

— bring it up top, then apply the rule:

⟹ $-\dfrac{1}{2x^{2}} + C$

Example Logarithmic and exponential

\displaystyle\int \dfrac{1}{x}\,dx

— the logarithmic rule:

⟹ $\ln|x| + C$

\displaystyle\int e^{x}\,dx

— base

e

, unchanged:

⟹ $e^{x} + C$

\displaystyle\int e^{3x}\,dx

— keep the function, divide by the coefficient

3

⟹ $\dfrac{e^{3x}}{3} + C$

\displaystyle\int 2^{x}\,dx

— base other than

e

, divide by

\ln 2

⟹ $\dfrac{2^{x}}{\ln 2} + C$

Example A sum, and the product trap

\displaystyle\int \big(x^{2} + 4x - 3\big)\,dx

— integrate term by term:

⟹ $\dfrac{x^{3}}{3} + 2x^{2} - 3x + C$

\displaystyle\int x \cdot x^{2}\,dx

— do not integrate each factor and multiply. Simplify:

x \cdot x^{2} = x^{3}

⟹ $\dfrac{x^{4}}{4} + C$

Example Definite integrals

Integrate first to get the antiderivative, then substitute the top limit minus the bottom limit.

\displaystyle\int_{0}^{2} 3x^{2}\,dx = \big[\,x^{3}\,\big]_{0}^{2} = 8 - 0

⟹ $8$

\displaystyle\int_{1}^{e} \dfrac{1}{x}\,dx = \big[\,\ln|x|\,\big]_{1}^{e} = 1 - 0

⟹ $1$

\displaystyle\int_{3}^{1} 2x\,dx = \big[\,x^{2}\,\big]_{3}^{1} = 1 - 9

— the limits run backwards, so the sign flips:

⟹ $-8$

Summary

Constant: $\displaystyle\int k\,dx = kx + C$ . Power rule: $\displaystyle\int x^{n}\,dx = \dfrac{x^{n+1}}{n+1} + C$ for $n \neq -1$ .
The power rule breaks at $n = -1$ ; the logarithmic rule fills the gap: $\displaystyle\int \dfrac{1}{x}\,dx = \ln|x| + C$ .
Exponentials: $\displaystyle\int e^{x}\,dx = e^{x} + C$ , $\displaystyle\int e^{kx}\,dx = \dfrac{e^{kx}}{k} + C$ , $\displaystyle\int a^{x}\,dx = \dfrac{a^{x}}{\ln a} + C$ .
Pull constants out; split sums and differences term by term.
Definite integral: $F(b) - F(a)$ ; reversing the limits negates the result.
Never split a product. Simplify first, or wait for integration by parts.

Mastering the Integration Rules