MP Board Class 12th Maths Book Solutions Chapter 7 समाकलन Ex 7.11

In this article, we share MP Board Class 12th Maths Book Solutions Chapter 7 समाकलन Ex 7.11 Pdf, These solutions are solved by subject experts from the latest MP Board books.

MP Board Class 12th Maths Book Solutions Chapter 7 समाकलन Ex 7.11

निश्चित समाकलनों में गुणधर्मों का उपयोग करते हुए 1 से 19 तक के प्रश्नों में समाकलनों का मान ज्ञात कीजिए-
प्रश्न 1.
\(\int_{0}^{\pi / 2} \cos ^{2} x d x\)
हल:
MP Board Class 12th Maths Book Solutions Chapter 7 समाकलन Ex 7.11 1

प्रश्न 2.
\(\int_{0}^{\pi / 2} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x\)
हल:
MP Board Class 12th Maths Book Solutions Chapter 7 समाकलन Ex 7.11 2
MP Board Class 12th Maths Book Solutions Chapter 7 समाकलन Ex 7.11 3

प्रश्न 3.
\(\int_{0}^{\pi / 2} \frac{\sin ^{3 / 2} x d x}{\sin ^{3 / 2} x+\cos ^{3 / 2} x} d x\)
हल:
MP Board Class 12th Maths Book Solutions Chapter 7 समाकलन Ex 7.11 4

प्रश्न 4.
\(\int_{0}^{\pi / 2} \frac{\cos ^{5} x}{\sin ^{5} x+\cos ^{5} x} d x\)
हल:
MP Board Class 12th Maths Book Solutions Chapter 7 समाकलन Ex 7.11 5

MP Board Solutions

प्रश्न 5.
\(\int_{-5}^{5}|x+2| d x\)
हल:
\(\int_{-5}^{5}|x+2| d x\)
MP Board Class 12th Maths Book Solutions Chapter 7 समाकलन Ex 7.11 6

प्रश्न 6.
\(\int_{2}^{8}|x-5| d x\)
हल:
MP Board Class 12th Maths Book Solutions Chapter 7 समाकलन Ex 7.11 7

प्रश्न 7.
\(\int_{0}^{1} x(1-x)^{n} d x\)
हल:
MP Board Class 12th Maths Book Solutions Chapter 7 समाकलन Ex 7.11 8
MP Board Class 12th Maths Book Solutions Chapter 7 समाकलन Ex 7.11 9

MP Board Solutions

प्रश्न 8.
\(\int_{0}^{\pi / 4} \log (1+\tan x) d x\)
हल:
MP Board Class 12th Maths Book Solutions Chapter 7 समाकलन Ex 7.11 10

प्रश्न 9.
\(\int_{0}^{2} x \sqrt{2-x} d x\)
हल:
MP Board Class 12th Maths Book Solutions Chapter 7 समाकलन Ex 7.11 11
MP Board Class 12th Maths Book Solutions Chapter 7 समाकलन Ex 7.11 12

प्रश्न 10.
\(\int_{0}^{\pi / 2}(2 \log \sin x-\log \sin 2 x) d x\)
हल:
MP Board Class 12th Maths Book Solutions Chapter 7 समाकलन Ex 7.11 13

प्रश्न 11.
\(\int_{-\pi / 2}^{\pi / 2} \sin ^{2} x d x\)
हल:
MP Board Class 12th Maths Book Solutions Chapter 7 समाकलन Ex 7.11 14
MP Board Class 12th Maths Book Solutions Chapter 7 समाकलन Ex 7.11 15

MP Board Solutions

प्रश्न 12.
\(\int_{0}^{\pi} \frac{x d x}{1+\sin x}\)
हल:
MP Board Class 12th Maths Book Solutions Chapter 7 समाकलन Ex 7.11 16

प्रश्न 13.
\(\int_{-\pi / 2}^{\pi / 2} \sin ^{7} x d x\)
हल:
माना \(\int_{-\pi / 2}^{\pi / 2} \sin ^{7} x d x\)
क्योंकि sin7 x एक विषम फलन है।
अतः I = 0.

प्रश्न 14.
\(\int_{0}^{2 \pi} \cos ^{5} x d x\)
हल:
माना I = \(\int_{0}^{2 \pi} \cos ^{5} x d x\)
माना f (x) = cos5 x तब
f(2π – x) = [cos (2π – x)]5 = cos5 x = f (x)
MP Board Class 12th Maths Book Solutions Chapter 7 समाकलन Ex 7.11 17

MP Board Solutions

प्रश्न 15.
\(\int_{0}^{\pi / 2} \frac{\sin x-\cos x}{1+\sin x \cos x} d x\)
हल:
MP Board Class 12th Maths Book Solutions Chapter 7 समाकलन Ex 7.11 18

प्रश्न 16.
\(\int_{0}^{\pi} \log (1+\cos x) d x\)
हल:
MP Board Class 12th Maths Book Solutions Chapter 7 समाकलन Ex 7.11 19
MP Board Class 12th Maths Book Solutions Chapter 7 समाकलन Ex 7.11 20

प्रश्न 17.
\(\int_{0}^{a} \frac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}} d x\)
हल:
MP Board Class 12th Maths Book Solutions Chapter 7 समाकलन Ex 7.11 21

प्रश्न 18.
\(\int_{0}^{4}|x-1| d x\)
हल:
\(\int_{0}^{4}|x-1| d x\)
MP Board Class 12th Maths Book Solutions Chapter 7 समाकलन Ex 7.11 22

MP Board Solutions

प्रश्न 19.
\(\int_{0}^{a} f(x) g(x) d x=2 \int_{0}^{a} f(x) d x\)
यदि f और g को f(x) = f(a – x) एवं g(x) + g(x – a) = 4 के रूप में परिभाषित किया गया है।
हल:
MP Board Class 12th Maths Book Solutions Chapter 7 समाकलन Ex 7.11 23

प्रश्न 20 एवं 21 में सही उत्तर का चयन कीजिए।
प्रश्न 20.
\(\int_{-\pi / 2}^{\pi / 2}\left(x^{3}+x \cos x+\tan ^{5} x+1\right) d x\) का मान है-
(A) 0
(B) 2
(C) π
(D) 1
हल:
MP Board Class 12th Maths Book Solutions Chapter 7 समाकलन Ex 7.11 24
MP Board Class 12th Maths Book Solutions Chapter 7 समाकलन Ex 7.11 25
अतः विकल्प (C) सही है।

प्रश्न 21.
\(\int_{0}^{\pi / 2} \log \left(\frac{4+3 \sin x}{4+3 \cos x}\right)\) का मान है
(A) 2
(B) \(\frac{3}{4}\)
(C) 0
(D) -2
हल:
MP Board Class 12th Maths Book Solutions Chapter 7 समाकलन Ex 7.11 26
अतः l = 0
अतः विकल्प (C) सही है।