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A particle moves on a straight line with velocity function $ v(t) = \sin \omega t \cos^2 \omega t $. Find its position function $ s = f(t) $ if $ f(0) = 0 $.

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 2

Trigonometric Integrals

Integration Techniques

University of Michigan - Ann Arbor

Idaho State University

Boston College

Lectures

01:53

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

27:53

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

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A particle moves on a stra…

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particle moves in a straig…

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A particle moves along a s…

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A particle moves along an …

02:06

a particle moves on a straight line with velocity function V and we were to find it position function and were given an initial condition over here after zero zero. So we know that velocities the derivative of the position. So we know that FT or S they're they're equal is the anti derivative of the and now we can go ahead and replace fee with a sign of omega T times co sign squared omega T Now for this problem, we should go ahead and use the u substitution take you to be co sign Omega team So that do you is by the chain rule negative omega sign of a mega t titi and we can rewrite this negative to you over omega equals sign of omega T TNT. So we have Let's simplify this. So we have a negative one over Omega Integral You square, do you, which is negative? One over omega You cubed over three plus e Unless he's our combats. Are u substitution to rewrite? This is negative. Cho sang cute Omega team over three Omega plus e So again, this is just using our substitution And now we confined See because of our initial condition. Half of zero is zero. Oops. So let's use that. So we know zero well equal, Negative, cosign Cubed of zero over three omega pussy. And we are given that this is zero. So this means that sea is cosign cubed of zero over three Omega and we know Coastline of zero was one. So we have won over three Omega. So this is our value for sea. So our final answer using using this fags in our value for sea begin apply both disease at the same time. And we have ten s equals negative. Cho sang cute omega t over three Omega plus he which was won over three Omega. And that's our answer for us.

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