RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.9

Rajasthan Board RBSE Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.9 Textbook Exercise Questions and Answers.

RBSE Class 12 Maths Solutions Chapter 7 Integrals Ex 7.9

Question 1.
\(\int_{-1}^{1}(x + 1)\) dx
Answer:

Question 2.
\(\int_{2}^{3} \frac{1}{x}\) dx
Answer:
Let

= log 3 - log 2 = log \(\frac{3}{2}\)

Question 3.
\(\int_{1}^{2}\) (4x³ - 5x² + 6x + 9) dx
Answer:

Question 4.
\(\int_{0}^{\pi / 4}\) sin 2x dx
Answer:

Question 5.
\(\int_{0}^{\pi / 2}\) cos 2x dx
Answer:

Question 6.
\(\int_{4}^{5}\) e^x dx
Answer:
Let I = \(\int_{4}^{5}\) e^x dx

= e⁵ - e⁴ = e⁴ (e - 1)

Question 7.
\(\int_{0}^{\pi / 4}\) tan x dx
Answer:

= (log sec\(\frac{\pi}{4}\) - log sec 0)
= (log √2 - log 1)
= log 2^1/2 - 0 = \(\frac{1}{2}\)log 2

Question 8.
\(\int_{\pi / 6}^{\pi / 4}\) cosec x dx
Answer:

Question 9.
\(\int_{0}^{1} \frac{d x}{\sqrt{1-x^{2}}}\)
Answer:

Question 10.
\(\int_{0}^{1} \frac{d x}{1+x^{2}}\)
Answer:

Question 11.
\(\int_{2}^{3} \frac{d x}{x^{2}-1}\)
Answer:

Question 12.
\(\int_{0}^{\pi / 2}\) cos² x dx
Answer:

Question 13.
\(\int_{2}^{3} \frac{x d x}{x^{2}+1}\)
Answer:
Let I = \(\int_{2}^{3} \frac{x d x}{x^{2}+1}\)
Putting x² + 1 = t
⇒ 2x dx = dt ⇒ x dx = \(\frac{1}{2}\) dt
When x = 3, then t = 3² + 1 = 10
When x = 2, then t = 2² + 1 = 5

Question 14.
\(\int_{0}^{1} \frac{2 x+3}{5 x^{2}+1}\) dx
Answer:

Question 15.
\(\int_{0}^{1}\) xe^x2
Answer:
Let I = \(\int_{2}^{3}\) xe^x2 dx
Putting x² = t
⇒ 2x dx = dt ⇒ x dx = \(\frac{1}{2}\) dt
When x = 1, then t = 1
When x = 0, then t = 0

Question 16.
\(\int_{1}^{2} \frac{5 x^{2}}{x^{2}+4 x+3}\) dx
Answer:

⇒ 20x + 15 = A(x + 1) + B(x + 3)
Puffing x = - 1
20(- 1) + 15 = A(- 1 + 1) + B(- 1 + 3)
⇒ - 5 = 0 + 2B
∴ B = - \(\frac{5}{2}\)
Putting x = - 3
20(- 3) + 15 = A(- 3 + 1) + B(- 3 + 3)
⇒ - 60 + 15 = - 2A
⇒ - 45 = - 2A
∴ A = \(\frac{45}{2}\)

Question 17.
\(\int_{0}^{\pi / 4}\) (2 sec² x + x³ + 2) dx
Answer:

Question 18.
\(\int_{0}^{\pi}\) (sin²\(\frac{x}{2}\) - cos²\(\frac{x}{2}\)) dx
Answer:

Question 19.
\(\int_{0}^{2} \frac{6 x+3}{x^{2}+4}\) dx
Answer:

Question 20.
\(\int_{0}^{1}\left(x e^{x}+\sin \frac{\pi x}{4}\right)\) dx
Answer:

Question 21.
\(\int_{1}^{\sqrt{3}} \frac{d x}{1+x^{2}}\) equals:
(A) \(\frac{\pi}{3}\)
(B) \(\frac{2 \pi}{3}\)
(C) \(\frac{\pi}{6}\)
(D) \(\frac{\pi}{12}\)
Answer:

Hence, (D) is the correct answer.

Question 22.
\(\int_{0}^{2 / 3} \frac{d x}{4+9 x^{2}}\) equals:
(A) \(\frac{\pi}{6}\)
(B) \(\frac{\pi}{12}\)
(C) \(\frac{\pi}{24}\)
(D) \(\frac{\pi}{4}\)
Answer:

Hence, (C) is the correct answer.