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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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In summary, substitution is a method used in calculus to simplify the evaluation of definite integrals by replacing the variable with a new one. It should be used when the integrand is complicated or contains trigonometric expressions or rational functions. The process involves choosing a suitable substitution, substituting the new variable, rewriting the limits of integration, and solving the integral. Some common substitutions include trigonometric, u-substitution, and inverse trigonometric substitutions. A successful substitution simplifies the integrand and the limits of integration, and the final answer should be in terms of the original variable and match results from other methods.
Why does the 3x^2 disappear? Doesn't it play a role in the answer?
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It is explained in the first line of blue text:
[tex]du = 3x^2 dx[/tex]
[tex]du = 3x^2 dx[/tex]
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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
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No, it's a simple substitution.
[tex]u = x^3 + 1[/tex]
Differentiating both sides with respect to x:
[tex]\frac{du}{dx} = 3x^2[/tex]
or equivalently
[tex]du = 3x^2 dx[/tex]
Now substitute into the original integral. The [itex]3x^2 dx[/itex] turns into [itex]du[/itex], and [itex]\sqrt{x^3 + 1}[/itex] becomes [itex]\sqrt{u}[/itex]. The endpoints of the integral also change accordingly.
[tex]u = x^3 + 1[/tex]
Differentiating both sides with respect to x:
[tex]\frac{du}{dx} = 3x^2[/tex]
or equivalently
[tex]du = 3x^2 dx[/tex]
Now substitute into the original integral. The [itex]3x^2 dx[/itex] turns into [itex]du[/itex], and [itex]\sqrt{x^3 + 1}[/itex] becomes [itex]\sqrt{u}[/itex]. The endpoints of the integral also change accordingly.
FAQ: Substitution of a definite integral
What is substitution in a definite integral?
Substitution, also known as change of variables, is a method used in calculus to simplify the evaluation of definite integrals. It involves replacing the variable in an integral with a new variable in order to simplify the integrand.
When should substitution be used in a definite integral?
Substitution should be used when the integrand contains a complicated expression that can be simplified by replacing the variable with a new one. It is also useful when the integrand contains a trigonometric expression or a rational function.
What is the process for substitution in a definite integral?
The process for substitution involves choosing a suitable substitution, substituting the new variable into the integral, rewriting the limits of integration in terms of the new variable, and then solving the integral using the new limits and variable. Finally, the answer is converted back to the original variable if necessary.
What are some common substitutions used in definite integrals?
Some common substitutions used in definite integrals include trigonometric substitutions, u-substitution, and inverse trigonometric substitutions. The choice of substitution will depend on the form of the integrand and the limits of integration.
How do I know if my substitution was successful in a definite integral?
If the substitution simplifies the integrand and the limits of integration can be rewritten in terms of the new variable, then it is considered a successful substitution. The final answer should also be in terms of the original variable and should match the results obtained using other methods.