Integral Toolbox Integral Toolbox

Introduction

Note the following properties of integrals:

The antiderivative of a constant and a function is equal to the product of the constant and the antiderivative of the function.

\int [ c \cdot f(x) ] \, dx = c \cdot \int f(x) \, dx
The antiderivative of two summed functions is equal to the sum of their antiderivatives.
\int [ f(x) + g(x) ] \, dx = \int f(x) \, dx + \int g(x) \, dx
The integral of a function on
(a,c)
is equal to the sum of the integral of the function on
(a,b)
and
(b,c)
.
\int_a^c f(x) \, dx = \int_a^b f(x) \, dx + \int_b^c f(x) \, dx
The integral of a function on
(a,c)
is equal to the negative of the integral of the function on
(c,a)
.
\int_a^c f(x) \, dx = -\int_c^a f(x) \, dx

Examples

1.

\begin{align*} \int 3x^2 + 6x \, dx &= 3 \int x^2 + 2x \, dx \\\\ &= 3 [\int x^2 \, dx + \int 2x \, dx] \\\\ &= 3 [\frac{x^3}{3} + x^2 + C] \\\\ &= x^3 + 3x^2 + C \end{align*}
2.
\begin{align*} \int_1^3 3x^2 + 6x \, dx &= \int_1^2 3x^2 + 6x \, dx + \int_2^3 3x^2 + 6x \, dx \\\\ &= [x^3 + 3x^2 + C] \Big|_1^2 + [x^3 + 3x^2 + C] \Big|_2^3 \\\\ &= [20 - 4] + [54 - 20] \\\\ &= 50 \end{align*}
3.
\begin{align*} \int_3^1 3x^2 + 6x \, dx &= -\int_1^3 3x^2 + 6x \, dx \\\\ &= -50 \end{align*}