   Chapter 4.R, Problem 40E

Chapter
Section
Textbook Problem

# Find the derivative of the function. y = ∫ 2 x 3 x + 1 sin ( t 4 )   d t

To determine

To find:

Derivative of thegiven function

Explanation

1) Concept:

i) First Fundamental Theorem of Calculus

If f is continuous on a, b, then he function g defined by

gx=axf(t)dt

is continuous on a, b and differentiable on (a, b), and g'x=f(x)

ii) Chain Rule:

If y=f(u) and u=g(x) are both differentiable functions, then

dydu=dydu·dudx

2) Property:

i) abf(x)=acf(x)dx+cbfxdx

ii) baf(x)dx=-abf(x)dx

3) Given:

y=2x3x+1sin(t4) dt

4) Calculation:

By using property(i),

2x3x+1sin(t4) dt  =2xcsin(t4)dt+c3x+1sin(t4)dt

where c is any constant

by using property (ii),

2xcsin(t4)dt=-c2xsin(t4)dt

So,

2x3x+1sin(t4) dt  =-c2xsin(t4)dt+c3x+1sin(t4)dt

Now evaluating the integrals separately,

For, y=-c2xsin(t4)dt

Substitute, u=2x

Differentiating with respect to x,

dudx=2

y=-c

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

#### Sketch the graphs of the equations in Exercises 512. xy=4

Finite Mathematics and Applied Calculus (MindTap Course List)

#### 15. Compute .

Mathematical Applications for the Management, Life, and Social Sciences

#### Given the Taylor Series , a Taylor series for ex/2 is:

Study Guide for Stewart's Multivariable Calculus, 8th 