Integrating ex and 1/x

Integrating ex and 1/x

Integration of e^x and 1/x

The Basics

• For any constant x, the integral of e^x dx will always yield e^x + C, where C is the constant of integration.
• This occurs because the derivative of e^x is e^x, making the integral simply e^x.

Specifics of e^x

• When integrating a function with e being raised to a power of a function, you must divide the result by the derivative of that function.
• This is expressed as ∫ e^(f(x)) dx = 1/f’(x) * e^(f(x)) + C

Integration of 1/x

• The integration of 1/x with respect to x yields **ln

  x

  + C**.

• It’s essential to understand the absolute value is included due to the natural logarithm function being undefined for negative values.

Specifics of 1/x

• When integrating a function where 1 is divided by a function, you must multiply the result by the derivative of that function.

• This is expressed as ∫ 1/f(x) dx = ln

  f(x)

  * f’(x) + C

Exponential and Logarithmic Integration

• When dealing with both exponentials and logarithms in integration, specific rules must be applied for correct integration.
• If you are integrating a function within the form ∫ (u’*e^u) dx = e^u + C, remember, u is not necessarily x but rather a function of x.

• Similarly, for a function within the form ∫ (u’/u) dx = ln

  u

  + C, remember that u is a function of x.

Remember, always include the constant of integration (+ C) in your solutions.