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Calculus

Indefinite Integral

As noted in the previous section, the infinite many antiderivatives (with different constants) of a given function \(f(x)\) is called an indefinite integral and is written \(\int f(x) dx \).

The indefinite integral of a function \(f(x) \) can be defined as the antiderivative \(F(x) \) plus any constant \(c\),

\[\int f(x)dx = F(x) + c\]

Theorems of Linearity

1. If \(f\) and \(g\) are are continuous functions, then 

\[\int \left(f(x) \pm g(x)\right) dx = \int f(x) dx \pm \int g(x) dx \]

2. If \(f\) is a continuous function and \(k\) is a real constant, then 

\[  \int kf(x) dx = k\int f(x) dx \]

Power Rule for Integration

To find the differential of a power of a function, we multiply by the power, subtract 1 from it, and multiply by the differential of the function.

\[ \qquad \frac{d}{dx} x^n = \frac{x^{n-1}}{n-1} \]

To find the integral, we reverse this procedure to get the power rule for integration:

\[ \qquad \int u^n du = \frac{u^{n+1}}{n+1} + c \qquad (n \neq -1) \]

Examples

Find the following integrals:

\[\int 3A\sqrt{A}-5A^2)dA \]

Solution:

Simplify square roots into fractional exponents and apply addition rules for integrals.

\[=\int 3A^{\frac{3}{2}}dA - \int 5A^2dA \]

Move constants outside integrals

\[=3\int A^{\frac{3}{2}}dA - 5 \int A^2dA \]

Integrate using power rule for integration and don't forget the CONSTANT

\[=3\frac{A^{\frac{5}{2}}}{{\frac{5}{2}}} - 5{\frac{A^3}{3}} + c \]

Simplify

\[= \frac{6A^{\frac{5}{2}}}{5} - \frac{5A^3}{3} + c \]

For more complex powers you can use a substitution before applying the power rule.

\[\int (x^2 -x)\left(x^3 - \frac{3}{2}x^2)^8\right) dx \]

Solution:

Substitute the values inside the bracket and find the derivative of the substitution

\[Let \, u =  x^3 - \frac{3}{2}x^2, du = (3x^2 - 3x)dx \]

Substitute u and du in for x and dx in original integral

\[\int \left(\underbrace{x^3 - \frac{3}{2}x^2}_\text{u})^8\right) \underbrace{(x^2 -x)dx}_\text{du} \]

\[= \int u^8du \]

\[= \frac{u^9}{9} + c\]

Substitute x values back 

\[=\frac{\left(x^3 - \frac{3}{2}x^2 \right)^9}{9} + c \]

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