7. Antiderivatives & Indefinite Integrals

104–107. Functions from derivatives Find the function f with the following properties.

h'(x) = (x⁴ -2) /(1 + x²) ; h (1) = -(2/3) 

{Use of Tech} Graphing general solutions Graph several functions that satisfy each of the following differential equations. Then find and graph the particular function that satisfies the given initial condition.

f'(x) = 3x + sinx; f(0) = 3

Particular antiderivatives For the following functions f, find the antiderivative F that satisfies the given condition.

f(x) = 8x³ + sin x; F(0) = 2

Particular antiderivatives For the following functions f, find the antiderivative F that satisfies the given condition.

f(v) = sec v tan v; F(0) = 2, -π/2 < v < π/2

Particular antiderivatives For the following functions f, find the antiderivative F that satisfies the given condition.

f(x) = (3y + 5)/y; F(1) = 3. y > 0

Initial Value Problems

Solve the initial value problems in Exercises 89–92.

d^3 r/dt^3 = - cos t; r''(0) = r'(0) = 0 , r(0) = -1

Initial Value Problems

Solve the initial value problems in Exercises 71–90.

dy/dx = 2x − 7, y(2) = 0

Initial Value Problems

Solve the initial value problems in Exercises 71–90.

dy/dx = 1/x² + x, x > 0; y(2) = 1

Initial Value Problems

Solve the initial value problems in Exercises 71–90.

dy/dx = 3x⁻²ᐟ³, y(−1) = −5

Initial Value Problems

Solve the initial value problems in Exercises 71–90.

ds/dt = 1 + cos t, s(0) = 4

Initial Value Problems

Solve the initial value problems in Exercises 71–90.

dr/dθ = −π sin (πθ), r(0) = 0

Initial Value Problems

Solve the initial value problems in Exercises 71–90.

dv/dt = (1/2)sec t tan t, v(0) = 1

Initial Value Problems

Solve the initial value problems in Exercises 71–90.

d³y/dx³ = 6; y″(0) = −8, y′(0) = 0, y(0) = 5

Initial Value Problems

Find the curve y = f(x) in the xy-plane that passes through the point (9,4) and whose slope at each point is 3√x.

Solve the initial value problems in Exercises 87 and 88.

87. dy/dx = 1 + 1/x, y(1) = 3