antiderivative

Question

1)Find f if f ' (x) = 2 cos x + ((3x^2+5)/( 1+x^2)) and f(0) = −1

2)A particle moves in a straight line and has acceleration given by a(t) = 12t + 5 sin t. Its initial velocity is v(0) = −3cm/s and its initial displacement is s(0) = 7cm. Find its position function .

Answer 1

Step 1:

1)

The derivative function is and .

Integration of derivative function results the function it self. 

Consider .

Integrate on  each side with respect to .

Integral formulae:

and .

.

We have .

Substitute in .

Substitute in .

.

Solution:

.

Answer 2

Step 1:

2)

A particle moves in a straight line.

Acceleration  of the  particle is and

Initial velocity is cm/s.

Initial displacement cm.

Anti derivative of the acceleration function is velocity.

.

Integrate on each side with respect to .

We have, .

Substitute in .

Substitute in .

.

Step 2:

Anti derivative of the velocity function is position function.

. 

Integrate on  each side with respect to .

Power rule of integration : and .

We have, .

Substitute in .

Substitute in .

.

Position function of the particle .

Solution:

Position function of the particle .