Basic Rules of Integration#
The Rules#
Integral of a Costant
For any real number \(k\),
The Power Rule
For any real number \(n\neq -1\),
The Constant Multiple Rule
For any real number \(k\),
The Sum Rule
Integral of \(e^x\)
For any real number \(a\neq 0\),
Integral of \(1/x\)
The Substitution Rule
If \(u=g(x)\) and \(du = g'(x) ~dx\) then
Example 1#
Integral of a constant
Compute \(\displaystyle \int 13 ~dz\).
Step 1: Notice the differential \(dz\).
This indicates that we are looking for a function of \(z\), \(f(z)\), such that \(f'(z) = 13\).
Step 2: Recall the formula for the integral of a constant.
Step 3: Apply the rule with \(k = 13\).
Therefore,
In other words, every function of \(z\) with derivative equal to \(13\) can be written in the form \(13z + C\) for some constant \(C\).
Check Our Work.
We can verify our answer by showing \(13\) is the derivative of \(13z\).
Example 2#
Integral of a power function
Compute \(\displaystyle \int x^7 ~dx\).
Step 1: Recall the power rule.
For any real number \(n\neq -1\),
Step 2: Apply the power rule with \(n = 7\).
Check Our Work.
We can verify our answer by showing \(x^7\) is the derivative of \(\dfrac{x^8}{8}\).
Example 3#
Integral of a power function with rational exponent
Compute \(\displaystyle \int \frac{1}{\sqrt{y}} ~dy\).
Step 1: Rewrite the integrand in the appropriate form to apply the power rule.
Step 2: Apply the power rule with \(n = -1/2\).
Check Our Work.
We can verify our answer by showing \(\dfrac{1}{\sqrt{y}}\) is the derivative of \(2\sqrt{y}\).
Example 4#
Integral of a constant multiple of a power function
Compute \(\displaystyle \int 4x^{7/3} ~dx\).
Step 1: Recall the constant multiple rule.
For any real number \(k\),
Step 2: Apply the constant multiple rule with \(k=4\).
Check Our Work.
We can verify our answer by showing \(4x^{7/3}\) is the derivative of \(\dfrac{6}{5} x^{10/3}\).
Example 5#
Integral of a sum of power functions
Compute \(\displaystyle \int 5t^3-\frac{10}{t^{6}}+4\sqrt{t} ~dt\).
Step 1: Recall the sum and difference rule.
Step 2: Apply the sum, difference and constant multiple rules.
Step 3: Integrate each term and then simplify.
Check Our Work.
We can verify our answer by showing \(5t^3-\dfrac{10}{t^{6}}+4\sqrt{t}\) is the derivative of \(\dfrac{5}{4}t^4 + 2t^{-5} + \dfrac{8}{3}t^{3/2}\).
Example 6#
Integral of a polynomial divided by a power function
Compute \(\displaystyle \int \frac{4x^9 - 15x^4 + 7x^3}{x^4} ~dx\).
Step 1: Rewrite the integrand as a sum.
Step 2: Apply the sum and constant multiple rules.
Step 3: Integrate each term and then simplify.
Example 7#
Integral of an exponential function
Compute \(\displaystyle \int e^{2x/5} ~dx\).
Step 1: Recall the formula for the integral of \(e^{ax}\) for \(a\neq 0\).
Step 2: Apply the formula for the integral of \(e^{ax}\) with \(a=2/5\).
Example 8#
Integral of a sum of functions
Compute \(\displaystyle \int 3e^{2x}+\frac{8}{x}+\frac{4}{x^3} ~dx\).
Step 1: Apply the sum and constant multiple rules.
Step 2: Integrate each term and then simplify.
Example 9#
Integral of a product of functions
Compute \(\displaystyle \int (e^{3x} + 1)(e^{-3x} - 1) ~dx\).