Learning Outcomes

Upon completing this lesson, you will be able to:

• Find the indefinite integrals of exponential functions, including the natural exponential function $e^x$ and other bases $b^x$.
• Find the indefinite integrals of the six basic trigonometric functions.
• Find the indefinite integral of the natural logarithmic function, $\ln(x)$, using integration by parts.
• Combine these new rules with the basic integration rules to solve more complex integrals.

Topics Covered

• Exponential Functions
• Trigonometric Functions
• Logarithmic Functions
• Special Cases

I. Integrating Exponential Functions

Just as exponential functions have special properties in differentiation, they also have simple and elegant rules for integration.

A. The Natural Exponential Function

The natural exponential function, $e^x$, is unique in that it is its own derivative. Consequently, it is also its own antiderivative.

$$ \int e^x ,dx = e^x + C $$

B. Exponential Functions with Other Bases

To integrate an exponential function with a different base, such as $b^x$, we recall that its derivative is $\frac{d}{dx}(b^x) = b^x \ln(b)$. To reverse this, we must divide by the constant factor $\ln(b)$.

$$ \int b^x ,dx = \frac{b^x}{\ln(b)} + C, \quad \text{for } b > 0, b \neq 1 $$

Example 1

Evaluate

$$ \int (3e^x + x^2) ,dx $$

Using the Sum Rule: $\int 3e^x ,dx + \int x^2 ,dx$

$$ 3 \int e^x ,dx + \int x^2 ,dx = 3e^x + \frac{x^3}{3} + C $$

Example 2

Evaluate

$$ \int 5^x ,dx $$

Using the rule for other bases, where $b=5$

$$ \frac{5^x}{\ln(5)} + C $$

II. Integrating Trigonometric Functions

The basic rules for integrating trigonometric functions are derived directly by reversing their corresponding derivative rules.

A. Integrals of Sine and Cosine

• Since $\frac{d}{dx}(\cos x) = -\sin x$, the integral of sine is negative cosine.

  $$ \int \sin(x) ,dx = -\cos(x) + C $$

• Since $\frac{d}{dx}(\sin x) = \cos x$, the integral of cosine is sine.

  $$ \int \cos(x) ,dx = \sin(x) + C $$

B. Integrals of Other Trigonometric Functions

The remaining rules are also found by reversing differentiation formulas:

$$ \int \sec^2(x) ,dx = \tan(x) + C $$

$$ \int \csc^2(x) ,dx = -\cot(x) + C $$