The Fundamental Theorem of Calculus

Question

1. Use Part 1 of the FTC to find the derivative of the function

a) y=∫ from 1 to 2x    (sin(t))/(t) dt

b) g(x)= ∫ sinx to 0  sqrt(t) dt

c) f(x)= ∫ sqrt(x) to x^2  ((e^(-t)+e^(t))/(t)) dt

 

2. Evaluate the definite integral. Give exact solutions.

a) ∫ 1 to 9   (x−(2/sqrt(x))^2

b) ∫ 0 to (pi/3)  secx(secx-tanx) dx

c) ∫ (pi/6) to (pi/3)  (cos2x)/(cos^(2)x) dx

 

Answer 1

1. (a)

Step 1 :

The fundamental theorem of calculus, part 1 :

If  is continuous on then the function is defined by

is continuous on and differentiable on , and .

Step 2 :

The equation is

Compare with

The function .

Thus, from the fundamental theorem of calculus, part 1

The derivative of the function is,

.

The derivative of the function .

Solution :

The derivative of the function .

Answer 2

1. (b)

Step 1 :

The fundamental theorem of calculus, part 1 :

If  is continuous on then the function is defined by

is continuous on and differentiable on , and .

Step 2 :

The equation is .

Apply definition of special definite integrals: .

.

.

Apply derivative on each side.

Substitute and .

.

.

The derivative of the function .

Solution :

The derivative of the function .

Answer 3

Step 1:

2. (a)

The integral is .

.

Solution :

.

Answer 4

Step 1:

2. (b)

The integral is .

.

Solution :

.

Answer 5

Step 1:

2. (c)

The integral is .

.

Solution :

.

Answer 6

1. (c)

Step 1 :

The fundamental theorem of calculus, part 1 :

If  is continuous on then the function is defined by

is continuous on and differentiable on , and .

Step 2 :

The equation is .

Apply Additive interval property: .

Apply definition of special definite integrals: .

Apply derivative on each side.

.

The derivative of the function .

Solution :

The derivative of the function .