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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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]
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.