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Question 1:

By using the properties of definite integrals, evaluate the integrals

Solution 1:

Adding (1) and (2), we obtain

Comments

Azhar M

Question 2:

By using the properties of definite integrals, evaluate the integrals

Solution 2:

Adding (1) and (2), we obtain

Comments

Azhar M

Question 3:

By using the properties of definite integrals, evaluate the integrals

Solution 3:

Adding (1) and (2), we obtain

Comments

Azhar M

Question 4:

By using the properties of definite integrals, evaluate the integrals

Solution 4:

Adding (1) and (2), we obtain

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Azhar M

Question 5:

By using the properties of definite integrals, evaluate the integrals

Solution 5:

It can be seen that (x + 2) ≤ 0 on [−5, −2] and (x + 2) ≥ 0 on [−2, 5].

Comments

Azhar M

Question 6:

By using the properties of definite integrals, evaluate the integrals

Solution 6:

It can be seen that (x − 5) ≤ 0 on [2, 5] and (x − 5) ≥ 0 on [5, 8].

Comments

Azhar M

Question 7:

By using the properties of definite integrals, evaluate the integrals

Solution 7:

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Azhar M

Question 8:

By using the properties of definite integrals, evaluate the integrals

Solution 8:

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Azhar M

Question 9:

By using the properties of definite integrals, evaluate the integrals

Solution 9:

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Azhar M

Question 10:

By using the properties of definite integrals, evaluate the integrals

Solution 10:

Adding (1) and (2), we obtain

Comments

Azhar M

Question 11:

By using the properties of definite integrals, evaluate the integrals

Solution 11:

As sin²(−x) = (sin (−x))² = (−sin x)² = sin²x, therefore, sin²x is an even function.

It is known that if f(x) is an even function, then

Comments

Azhar M

Question 12:

By using the properties of definite integrals, evaluate the integrals

Solution 12:

Adding (1) and (2), we obtain

Comments

Azhar M

Question 13:

By using the properties of definite integrals, evaluate the integrals

Solution 13:

As sin⁷(−x) = (sin (−x))⁷ = (−sin x)⁷ = −sin⁷x, therefore, sin²x is an odd function.

It is known that, if f(x) is an odd function, then

Comments

Azhar M

Question 14:

By using the properties of definite integrals, evaluate the integrals

Solution 14:

It is known that,

Comments

Azhar M

Question 15:

By using the properties of definite integrals, evaluate the integrals

Solution 15:

Adding (1) and (2), we obtain

Comments

Azhar M

Question 16:

By using the properties of definite integrals, evaluate the integrals

Solution 16:

Adding (1) and (2), we obtain

sin (π − x) = sin x

Adding (4) and (5), we obtain

Let 2x = t ⇒ 2dx = dt

When x = 0, t = 0
and when $x = \frac{π}{2}, t = π$
∴ $I = \frac{1}{2} \int_{0}^{π} \log \sin t d t - \frac{π}{2} \log 2$
$\Rightarrow I = \frac{I}{2} - \frac{π}{2} \log 2 [from 3]$
$\Rightarrow \frac{I}{2} = - \frac{π}{2} \log 2$
$\Rightarrow I = - πlog 2$

Comments

Azhar M

Question 17:

By using the properties of definite integrals, evaluate the integrals

Solution 17:

It is known that,

Adding (1) and (2), we obtain

Comments

Azhar M

Question 18:

By using the properties of definite integrals, evaluate the integrals

Solution 18:

It can be seen that, (x − 1) ≤ 0 when 0 ≤ x ≤ 1 and (x − 1) ≥ 0 when 1 ≤ x ≤ 4

Comments

Azhar M

Question 19:

Show that if f and g are defined as and

Solution 19:

Adding (1) and (2), we obtain

Comments

Azhar M

Question 20:

The value of is

A. 0

B. 2

C. π

Solution 20:

It is known that if f(x) is an even function, then and

if f(x) is an odd function, then

Hence, the correct answer is C.

Comments

Azhar M

Question 21:

The value of is

A. 2

C. 0

Solution 21:

Adding (1) and (2), we obtain

Hence, the correct answer is C.

Comments

Tasneem

All Exercises - Chapter 7 - Integrals

Exercise 7.11

Miscellaneous Exercise 7

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Learn Exercise 7.11 with Free Lessons & Tips

All Exercises - Chapter 7 - Integrals

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