Problem 51
Integrating Inverse Functions Assume that the function \(f\) has an inverse. (a) Show that \(\int f^{-1}(x) d x=\int y f^{\prime}(y) d y .\) (Hint: Use the substitution \(y=f^{-1}(x) . )\) (b) Use integration by parts on the second integral in part (a) to show that $$\int f^{-1}(x) d x=\int y f^{\prime}(y) d y=x f^{-1}(x)-\int f(y) d y$$
Problem 51
In Exercises \(47-52,\) use the given trigonometric identity to set up a \(u\) -substitution and then evaluate the indefinite integral. $$\int \tan ^{4} x d x, \quad \tan ^{2} x=\sec ^{2} x-1$$
Problem 52
Integrating Inverse Functions Assume that the function \(f\) has an inverse. Use integration by parts directly to show that $$\int f^{-1}(x) d x=x f^{-1}(x)-\int x\left(\frac{d}{d x} f^{-1}(x)\right) d x$$
Problem 52
In Exercises \(47-52,\) use the given trigonometric identity to set up a \(u\) -substitution and then evaluate the indefinite integral. $$\int\left(\cos ^{4} x-\sin ^{4} x\right) d x, \quad \cos 2 x=\cos ^{2} x-\sin ^{2} x$$
Problem 53
Percentage Error Let \(y=f(x)\) be the solution to the initial value problem \(d y / d x=2 x+1\) such that \(f(1)=3 .\) Find the per- centage error if Euler's Method with \(\Delta x=0.1\) is used to approxi- mate \(f(1.4) .\)
Problem 53
In Exercises \(53-66,\) make a \(u\) -substitution and integrate from \(u(a)\) to \(u(b) .\) $$\int_{0}^{3} \sqrt{y+1} d y$$
Problem 53
Resistance Proportional to Velocity It is reasonable to assume that the air resistance encountered by a moving object, such as a car coasting to a stop, is proportional to the object's velocity. The resisting force on an object of mass \(m\) moving with velocity \(v\) is thus \(-k v\) for some positive constant \(k\). (a) Use the law Force = Mass \(\times\) Acceleration to show that the velocity of an object slowed by air resistance (and no other forces) satisfies the differential equation $$m \frac{d y}{d t}=-k v$$ (b) Solve the differential equation to show that \(v=v_{0} e^{-(k / m) t}\) where \(v_{0}\) is the velocity of the object at time \(t=0 . (c) If \)k$ is the same for two objects of different masses, which one will slow to half its starting velocity in the shortest time? Justify your answer.
Problem 54
Coasting to a Stop Assume that the resistance encountered by a moving object is proportional to the object's velocity so that its velocity is \(v=v_{0} e^{-(k / m) t} .\) (a) Integrate the velocity function with respect to \(t\) to obtain the distance function \(s .\) Assume that \(s(0)=0\) and show that $$s(t)=\frac{v_{0} m}{k}\left(1-e^{-(k / m) t}\right).$$ (b) Show that the total coasting distance traveled by the object as it coasts to a complete stop is \(v_{0} m / k .\)
Problem 54
In Exercises \(53-66,\) make a \(u\) -substitution and integrate from \(u(a)\) to \(u(b) .\) $$\int_{0}^{1} r \sqrt{1-r^{2}} d r$$
Problem 54
Percentage Error Let \(y=f(x)\) be solution to the initial value problem \(d y / d x=2 x-1\) such that \(f(2)=3 .\) Find the per- centage error if Euler's Method with \(\Delta x=-0.1\) is used to ap- proximate \(f(1.6) .\)
