How to integrate exponential functions : ExamSolutions Maths Revision Tutorials

Summary

This tutorial explains how to integrate exponential functions, starting with basic examples like y = e^x and progressing to more complex forms. It covers key concepts like integrating constants with exponential functions, applying the chain rule for functions like e^(ax + b), and handling fractional coefficients or exponential terms in the denominator. Practical examples are provided, including step-by-step integration for various forms, demonstrating how to handle constants, powers, and coefficients in integration. The tutorial concludes with a challenge for viewers to practice and solidify their understanding.

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Q & A

What is the integral of e^x with respect to x?

The integral of e^x with respect to x is e^x + C, where C is the constant of integration.

How does the presence of a constant in front of e^x affect its integration?

When a constant 'a' is multiplied by e^x, the integral becomes a * e^x + C. The constant is carried through the integration.

What happens when you integrate a function like 27e^x?

For the integral of 27e^x, the constant 27 remains in the result. The integral becomes 27e^x + C.

How do you integrate a function of the form e^(ax + b)?

To integrate e^(ax + b), apply the chain rule. The integral is e^(ax + b) / a + C, where 'a' is the coefficient of x and 'b' is a constant.

What is the general rule for integrating a * e^(ax + b)?

The general rule for integrating a * e^(ax + b) is (a / a) * e^(ax + b) + C, simplifying to e^(ax + b) + C.

How would you handle integrating an expression like 3/5e^(2x)?

For the integral of 3/5e^(2x), treat 3/5 as a constant. The result would be (3/10)e^(2x) + C.

How do you handle integration when the exponential term appears in the denominator?

When the exponential term appears in the denominator, rewrite it with a negative exponent. For example, 1/e^(5x) becomes e^(-5x), and then you integrate normally.

What happens when you integrate a function like 4/3e^(-5x)?

For the integral of 4/3e^(-5x), rewrite it as 4/3 * e^(-5x). The integral becomes (4/15)e^(-5x) + C.

How do you deal with integration when the exponential term has a negative exponent in the denominator?

If the exponential term is in the denominator with a negative exponent, you bring it up to the numerator as a positive exponent and proceed with the integration.

What should you do when integrating a more complex exponential function, like 2/3e^(3x - 2)?

For the integral of 2/3e^(3x - 2), treat 2/3 as a constant. Using the rule for integrating e^(ax + b), the result is (2/9)e^(3x - 2) + C.