Chapter 4: Integration
Section 4.3: Fundamental Theorem of Calculus and the Indefinite Integral
Example 4.3.5
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
Part (a)
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 = ⅆⅆxyxfyx
Solution from first principles
In the definition of w, set yx equal to v. This defines wvx.
wv=∫avft ⅆt
Apply the Chain rule to differentiate wvx with respect to x.
Obtain dwdv by the FTC.
w′x
=dwdvdvdx
=fv⋅y′x
=f(yx⋅y′x
Part (b)
Since wx=∫yxaft ⅆt= −∫ayxft ⅆt, by the results of Part (a), it follows that w′x=−fyxy′x
Part (c)
Control-drag the integral. Context Panel: Differentiate≻With Respect To≻x
∫0xsinht ⅆt→differentiate w.r.t. x12sinhxx
Apply the result of Part (a)
Evaluate the integrand at the upper limit and multiply by the derivative of the upper limit.
ddx∫0xsinht ⅆt
=sinhx⋅ddxx
=sinhx 12x
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