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bobsmith76
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Homework Statement
Why does the 3x^2 disappear? Doesn't it play a role in the answer?
Substitution of a definite integral
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SUMMARY
The discussion clarifies the process of substitution in definite integrals, specifically using the integral of the function involving the substitution u = x^3 + 1. The derivative du = 3x^2 dx is highlighted, demonstrating that the 3x^2 term is replaced by du during the substitution process. This transformation simplifies the integral, allowing \sqrt{x^3 + 1} to be expressed as \sqrt{u}. The endpoints of the integral are also adjusted to reflect the new variable.
PREREQUISITES- Understanding of definite integrals
- Familiarity with substitution methods in calculus
- Knowledge of differentiation and its application
- Basic algebraic manipulation skills
- Study the process of variable substitution in integrals
- Learn about the Fundamental Theorem of Calculus
- Explore advanced integration techniques such as integration by parts
- Practice solving definite integrals with various substitution methods
Students studying calculus, educators teaching integration techniques, and anyone seeking to improve their understanding of definite integrals and substitution methods.
Why does the 3x^2 disappear? Doesn't it play a role in the answer?
- #2
jbunniii
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It is explained in the first line of blue text:
du = 3x^2 dx
du = 3x^2 dx
- #3
bobsmith76
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I think after you find the derivative of u, which is 3x^2, you divide that by 3x^2 which is 1, but I'm not sure
- #4
jbunniii
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No, it's a simple substitution.
u = x^3 + 1
Differentiating both sides with respect to x:
\frac{du}{dx} = 3x^2
or equivalently
du = 3x^2 dx
Now substitute into the original integral. The 3x^2 dx turns into du, and \sqrt{x^3 + 1} becomes \sqrt{u}. The endpoints of the integral also change accordingly.
u = x^3 + 1
Differentiating both sides with respect to x:
\frac{du}{dx} = 3x^2
or equivalently
du = 3x^2 dx
Now substitute into the original integral. The 3x^2 dx turns into du, and \sqrt{x^3 + 1} becomes \sqrt{u}. The endpoints of the integral also change accordingly.