How Do You Integrate Functions Involving Powers and Trigonometric Terms?

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SUMMARY

This discussion focuses on integrating functions involving powers and trigonometric terms, specifically the integral of x^(1/3) divided by (x^(1/3) - 1) and finding the area bounded by the curve y = 2x - tan(0.3x) between x = 1 and x = 4. The integration process involves substitution with u = x^(1/3), leading to the integral of u^3/(u-1) and further decomposition into simpler integrals. Additionally, participants confirm that the area can be found by integrating the function 2x - tan(0.3x) over the specified interval.

PREREQUISITES
  • Understanding of integral calculus
  • Familiarity with substitution methods in integration
  • Knowledge of trigonometric functions and their properties
  • Ability to evaluate definite integrals
NEXT STEPS
  • Study integration techniques involving substitution and partial fractions
  • Learn about the properties of trigonometric functions in calculus
  • Explore the application of definite integrals in finding areas under curves
  • Review advanced integration techniques, such as integration by parts
USEFUL FOR

Students and educators in calculus, mathematicians focusing on integration techniques, and anyone interested in solving complex integrals involving powers and trigonometric functions.

Blade
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INT x^(1/3) / x^(1/3) -1 dx
u = x^(1/3)
x = u^3
dx = 3u^2du

INT (u/u-1)3u^2du = 3
INT u^3/(u-1) du
INT [u^2 + u + 1/(u-1)]du
Then just integrate each part?

-----------
Find the area of the region bounded.

y=2x - tan(0.3x), x-1, x=4, y=0
Need some guidance where to start here, and how the conditions will be used.
 
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Looks reasonable to me!

What guidance do you need for the second problem?

Just integrate 2x- tan(0.3x) from 1 to 4.
 
Originally posted by Blade
INT x^(1/3) / x^(1/3) -1 dx
u = x^(1/3)
x = u^3
dx = 3u^2du

INT (u/u-1)3u^2du = 3
INT u^3/(u-1) du
INT [u^2 + u + 1/(u-1)]du
Then just integrate each part?

-----------
Find the area of the region bounded.

y=2x - tan(0.3x), x-1, x=4, y=0
Need some guidance where to start here, and how the conditions will be used.

I don't see any ln in the integrand.
 

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