Chapter 4: Q. 51 (page 373)

Use the Fundamental Theorem of Calculus to find the exact
values of each of the definite integrals in Exercises 19–64. Use
a graph to check your answer. (Hint: The integrands that involve
absolute values will have to be considered piecewise.)
$\int_{0}^{\frac{π}{2}} \sin x (1 + \cos x) d x$

Short Answer

Expert verified

$\int_{0}^{\frac{π}{2}} \sin x (1 + \cos x) d x = \frac{3}{2}$ .

Step by step solution

Step 1. Given information.

A definite integral is given as $\int_{0}^{\frac{π}{2}} \sin x (1 + \cos x) d x$ .

Step 2. Using the Fundamental theorem of Calculus.

We get

$\int_{0}^{\frac{π}{2}} \sin x (1 + \cos x) d x = \int_{0}^{\frac{π}{2}} \sin x d x + \int_{0}^{\frac{π}{2}} \sin x \cos x d x = {[- \cos x]}_{0}^{\frac{π}{2}} + \frac{1}{2} \int_{0}^{\frac{π}{2}} 2 \sin x \cos x d x = - \cos \frac{π}{2} + \cos 0 + \frac{1}{2} \int_{0}^{\frac{π}{2}} \sin (2 x) d x = - 0 + 1 + \frac{1}{2} {[- \frac{1}{2} \cos (2 x)]}_{0}^{\frac{π}{2}} = 1 - \frac{1}{4} [cosπ - \cos 0] = 1 - \frac{1}{4} (- 1 - 1) = 1 + \frac{1}{2} = \frac{3}{2}$

So the exact value of the given definite integral is $\frac{3}{2}$ .

Step 3. The graph to verify the answer is

The solution is area under graph which is

$a = 1.5 = \frac{3}{2}$

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Use the Fundamental Theorem of Calculus to find the exact
values of each of the definite integrals in Exercises 19–64. Use
a graph to check your answer. (Hint: The integrands that involve
absolute values will have to be considered piecewise.)
$\int_{0}^{\frac{π}{2}} \sin x (1 + \cos x) d x$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Using the Fundamental theorem of Calculus.

Step 3. The graph to verify the answer is

One App. One Place for Learning.

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Use the Fundamental Theorem of Calculus to find the exactvalues of each of the definite integrals in Exercises 19–64. Usea graph to check your answer. (Hint: The integrands that involveabsolute values will have to be considered piecewise.)∫0π2sinx(1+cosx)dx

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Using the Fundamental theorem of Calculus.

Step 3. The graph to verify the answer is

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Calculus

Theoretical and Mathematical Physics

Decision Maths

Logic and Functions

Pure Maths

Study anywhere. Anytime. Across all devices.

Use the Fundamental Theorem of Calculus to find the exact
values of each of the definite integrals in Exercises 19–64. Use
a graph to check your answer. (Hint: The integrands that involve
absolute values will have to be considered piecewise.)
$\int_{0}^{\frac{π}{2}} \sin x (1 + \cos x) d x$