The function e^∛x is known as the exponential function, which is a mathematical function that describes the growth of a quantity over time. It is often used in integration to model various real-world phenomena such as population growth, interest rates, and radioactive decay.
To solve the integration of e^∛x, you can use the substitution method by letting u = ∛x. This will transform the integral into a simpler form that can be easily solved. After solving the integral, don't forget to substitute back u = ∛x to get the final answer.
Yes, there is a specific rule for integrating e^∛x, which is the power rule for integration. This rule states that the integral of a function raised to a power can be found by adding 1 to the power and dividing the result by the new power. In this case, the integral of e^∛x is equal to e^(∛x+1)/(∛x+1).
Sure, here is a step-by-step guide for solving the integration of e^∛x using the substitution method:
Step 1: Let u = ∛x
Step 2: Calculate du/dx by differentiating both sides of the equation. In this case, du/dx = 1/(3∛x^2)
Step 3: Substitute u and du/dx into the integral, which becomes ∫e^u * 1/(3∛x^2) dx
Step 4: Rewrite the integral in terms of u, which becomes ∫e^u * 1/(3u^2) du
Step 5: Solve the integral using the power rule, which gives us the final answer of e^(∛x+1)/(∛x+1) + C
Integrating e^∛x has many applications in various fields of science and engineering. Some examples include modeling population growth, predicting the spread of diseases, calculating compound interest, and determining the half-life of radioactive substances. It is also used in physics and chemistry to solve problems involving exponential decay and growth.