How to find integrals of parent functions without any horizontal/vertical shift?

In summary, the conversation discusses how to find integrals of parent functions without any horizontal or vertical shift. It is mentioned that this can be done using a calculator that can only add, subtract, multiply, and divide. There is also a general formula for finding the integral of a function of the form Cxp where p ≠ -1. The conversation also addresses the difficulty of computing accurate approximations for 9^{1/7} and 14^{8/7} using a limited calculator. One suggested approach is to use polynomial expansion or iteration to find corrections.
  • #1
PeaceMartian
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TL;DR Summary: How to find integrals of parent functions without any horizontal/vertical shift?

Say you were given the equation :
Screenshot 2023-05-27 170512.png

How would you find :
Screenshot 2023-05-27 170938.png
with a calculator that can only add, subtract, multiply, divide

Is there a general formula?
 
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  • #2
You have a function of the form Cxp where p ≠ -1.
The integral is Cxp+1/(p+1)
 
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  • #3
The difficulty here is to compute (adequate approximations to) [itex]9^{1/7}[/itex] and [itex]14^{8/7} = 14 \cdot 14^{1/7}[/itex] using "a calculator that can only add, subtract, multiply, divide".
 
  • #4
pasmith said:
The difficulty here is to compute (adequate approximations to) [itex]9^{1/7}[/itex] and [itex]14^{8/7} = 14 \cdot 14^{1/7}[/itex] using "a calculator that can only add, subtract, multiply, divide".
1261/7≈1281/7=2
One could then do the polynomial expansion (2-x)7=126 and throw away higher terms to find corrections. Or do some iteration.
 

FAQ: How to find integrals of parent functions without any horizontal/vertical shift?

What is the integral of a constant function?

The integral of a constant function \( f(x) = c \) is given by \( \int c \, dx = cx + C \), where \( C \) is the constant of integration.

How do you find the integral of a linear function?

The integral of a linear function \( f(x) = ax + b \) is \( \int (ax + b) \, dx = \frac{a}{2}x^2 + bx + C \), where \( C \) is the constant of integration.

What is the integral of a power function?

The integral of a power function \( f(x) = x^n \) (where \( n \neq -1 \)) is \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.

How do you find the integral of an exponential function?

The integral of an exponential function \( f(x) = e^x \) is \( \int e^x \, dx = e^x + C \), where \( C \) is the constant of integration.

What is the integral of a trigonometric function like sine or cosine?

The integral of the sine function \( f(x) = \sin(x) \) is \( \int \sin(x) \, dx = -\cos(x) + C \), and the integral of the cosine function \( f(x) = \cos(x) \) is \( \int \cos(x) \, dx = \sin(x) + C \), where \( C \) is the constant of integration.

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