Chapter 4: Integration
Section 4.3: Fundamental Theorem of Calculus and the Indefinite Integral
If wx=∫ayxft ⅆt, show that w′x=fyx y′x.
If wx=∫yxaft ⅆt, show that w′x=−fyxy′x
Obtain ddx∫0xsinht ⅆt.
Solution via Maple
Control-drag the definition wx=…
Context Panel: Assign Function
wx=∫ayxft ⅆt→assign as functionw
Write the derivative w′x
Context Panel: Evaluate and Display Inline
w′x = ⅆⅆx⁢y⁡x⁢f⁡y⁡x
Solution from first principles
In the definition of w, set yx equal to v.
This defines wvx.
Apply the Chain rule to differentiate wvx with respect to x.
Obtain dwdv by the FTC.
Since wx=∫yxaft ⅆt= −∫ayxft ⅆt, by the results of Part (a), it follows that w′x=−fyxy′x
Control-drag the integral.
Context Panel: Differentiate≻With Respect To≻x
∫0xsinht ⅆt→differentiate w.r.t. x12⁢sinh⁡xx
Apply the result of Part (a)
Evaluate the integrand at the upper limit and multiply by the derivative of the upper limit.
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