NCERT Solutions for Class 12 Maths Chapter 7 समाकलन Ex 7.1

These NCERT Solutions for Class 12 Maths Chapter 7 समाकलन Ex 7.1 Questions and Answers are prepared by our highly skilled subject experts.

NCERT Solutions for Class 12 Maths Chapter 7 समाकलन Ex 7.1

प्रश्न 1.
sin 2x
हल:
∫sin 2x dx
हम जानते हैं कि
\(\frac{d}{d x}\) cos 2x = – 2 sin2x
⇒ \(\frac{d}{d x}\) (- \(\frac{1}{2}\)cos 2x) = sin2x
∴ ∫sin 2x dx = – \(\frac{1}{2}\)cos 2x + c

प्रश्न 2.
cos 3.x
हल:
∫cos3x dx
हम जानते हैं कि
\(\frac{d}{d x}\) (sin 3x) = 3 cos 3x
⇒ \(\frac{d}{d x}\) (\(\frac{1}{3}\) sin3x) = cos 3x
∴ ∫cos 3x dx = \(\frac{1}{3}\) sin 3x + C

प्रश्न 3.
e^2x
हल:
∫e^2x dx
हम जानते हैं कि
\(\frac{d}{d x}\) (e^2x) = 2e^2x
⇒ \(\frac{d}{d x}\) (\(\frac{1}{2}\) e^2x) = e^2x
∴ ∫e^2xdx = \(\frac{1}{2}\) e^2x + C

प्रश्न 4.
(ax + b)²
हल:
∫(ax + b)² dx
हम जानते हैं कि
\(\frac{d}{d x}\) (ax + b)³ = 3a(ax + b)²
\(\frac{d}{d x}\) (\(\frac{1}{3a}\) (ax + b)³) = (ax + b)²
∴ ∫(ax + b)² dx = \(\frac{1}{3a}\) (ax + b)³ + C

प्रश्न 5.
sin 2x – 4e^3x
हल:
∫(sin 2x – 4e^3x) dx
= ∫sin 2x dx – 4∫e^3x dx ……….(1)
∵ \(\frac{d}{d x}\) cos 2x = – 2 sin 2x
⇒ \(\frac{d}{d x}\)(-\(\frac{1}{2}\) cos2x) = sin2x
⇒ ∫sin2x dx = – \(\frac{1}{2}\) cos 2x + C₁ …………..(2)
पुनः \(\frac{d}{d x}\) e^3x = 3e^3x
⇒ \(\frac{d}{d x}\)(\(\frac{1}{3}\)e^3x) = e^3x
⇒ ∫e^3x = \(\frac{1}{3}\)e^3x + C₂ ………..(3)
(1), (2) तथा (3) से,
∫(sin 2x – 4e^3x) dx = –\(\frac{1}{2}\) cos 2x – \(\frac{4}{3}\)e^3x + C
(: C₁ + C₂ = C)

निम्नलिखित समाकलनों को ज्ञात कीजिए।
प्रश्न 6.
∫(4e^3x + 1) dx
हल:
∫(4e^3x + 1) dx = 4∫e^3x dx + ∫1 dx
= 4. \(\frac{e^{3 x}}{3}\) + x + C
= \(\frac{4}{3}\)e^3x + x + C
∵[\(\frac{d}{d x}\)e^ax = ae^ax
∴ \(\frac{d}{d x}\)(\(\frac{e^{a x}}{a}\)) = e^ax
⇒ ∫e^3x dx = \(\frac{e^{a x}}{a}\)

प्रश्न 7.
∫x² (1 – \(\frac{1}{x^2}\)) dx
हल:
∫x² (1 – \(\frac{1}{x^2}\)) dx = ∫(x² – 1) dx
= ∫x² dx – ∫1 dx
= \(\frac{x^3}{3}\) – x + C

प्रश्न 8.
∫(ax² + bx + c) dx
हल:
∫(ax² + bx + c) dx
= a∫x² dx + b∫x dx + c∫1.dx
= a\(\frac{x^{2+1}}{2+1}\) + b\(\frac{x^{1+1}}{1+1}\) + cx + C
= a\(\frac{x^3}{3}\) + b\(\frac{x^2}{2}\) + cx + C
= \(\frac{a x^3}{3}\) + \(\frac{b x^2}{2}\) + cx + C

प्रश्न 9.
∫(2x² + e^x) dx
हल:
∫(2x² + e^x) dx
= 2∫x² dx + ∫e^x dx
= 2 \(\frac{x^{2+1}}{2+1}\) + e^x + C
= \(\frac{2}{3}\) x³ + e^x + C

प्रश्न 10.
∫(√x + \(\frac{1}{\sqrt{x}}\))² dx
हल:

प्रश्न 11.
\(\int \frac{x^3+5 x^2-4}{x^2}\) dx
हल:
\(\int \frac{x^3+5 x^2-4}{x^2}\) dx
= ∫(\(\frac{x^3}{x^2}\) + \(\frac{5 x^2}{x^2}\) – \(\frac{4}{x^2}\))dx
= ∫x dx + 5∫1.dx – 4∫x^-2 dx
= \(\frac{x^2}{2}\) + 5x – 4. \(\frac{x^{-2+1}}{-2+1}\) + C
= \(\frac{x^2}{2}\) + 5x + \(\frac{4}{x}\) + C

प्रश्न 12.
\(\int \frac{x^3+3 x+4}{\sqrt{x}}\) dx
हल:

प्रश्न 13.
\(\int \frac{x^3-x^2+x-1}{x-1}\) dx
हल:
\(\int \frac{x^3-x^2+x-1}{x-1}\) dx
= \(\int \frac{x^2(x-1)+1(x-1)}{(x-1)}\)dx
= \(\int \frac{\left(x^2+1\right)(x-1)}{(x-1)}\)dx
= ∫(x² + 1)dx = ∫x²dx + ∫1.dx
= \(\frac{x^3}{3}\) + x + C

प्रश्न 14.
∫(1 – x)√x dx
हल:
∫(1 – x)√x dx
= ∫(√x – x√x)dx
= ∫(x^1/2 – x.x^1/2)dx
= ∫x^1/2dx – ∫x^3/2dx
= \(\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}\) – \(\frac{x^{\frac{3}{2}+1}}{\frac{3}{2}+1}\) + C
= \(\frac{x^{3 / 2}}{\frac{3}{2}}\) – \(\frac{x^{5 / 2}}{\frac{5}{2}}\) + C
= \(\frac{2}{3}\) x^3/2 – \(\frac{2}{5}\) x^5/2 + C

प्रश्न 15.
∫√x(3x² + 2x + 3)dx
हल:

प्रश्न 16.
∫(2x – 3 cos x + e^x) dx
हल:
∫(2x – 3 cos x + e^x) dx
= 2∫x dx – 3∫cos x dx + ∫e^x dx
= 2.\(\frac{x^2}{2}\) – 3sin x + e^x + C
= x² – 3 sin x + e^x + C

प्रश्न 17.
∫(2x² – 3 sin x + 5√x) dx
हल:
∫(2x² – 3 sin x + 5√x) dx
= 2∫x² dx – 3 ∫sin x dx + 5 ∫x^1/2 dx
= 2. \(\frac{x^3}{3}\) – 3 (-cos x) + 5 \(\frac{x^{3 / 2}}{3 / 2}\) + C
= \(\frac{2}{3}\) x³ + 3cosx + \(\frac{10}{3}\) x^3/2 + C

प्रश्न 18.
∫sec x (sec x + tan x) dx
हल:
∫secx (secx + tan x) dx
= ∫sec² x dx + ∫sec x tan x dx
= tan x + secx + C
[∵ sec² x dx = tan x
∫secx tan x dx = sec x]

प्रश्न 19.

हल:

= ∫\(\frac{\sin ^2 x}{\cos ^2 x}\) dx
= ∫tan² x dx
= ∫(sec²x – 1)dx
= ∫sec²xdx – ∫1.dx
= tan x – x + C

प्रश्न 20.
∫\(\frac{2-3 \sin x}{\cos ^2 x}\) dx
हल:
∫\(\frac{2-3 \sin x}{\cos ^2 x}\) dx
= ∫\(\frac{2}{\cos ^2 x}\) dx – 3∫\(\frac{\sin x}{\cos ^2 x}\) dx
= ∫sec²x dx – 3∫secxtanx dx
= 2 tan x – 3 secx + C

प्रश्न 21 एवं 22 में सही उत्तर का चयन कीजिए:
प्रश्न 21.
(√x + \(\frac{1}{\sqrt{x}}\)) का प्रतिअवकलज है:
(A) \(\frac{1}{3}\) x^1/3 + 2x^1/2 + C
(B) \(\frac{2}{3}\) x^2/3 + \(\frac{1}{2}\)x² + C
(C) \(\frac{2}{3}\) x^3/2 + 2x^1/2 + C
(D) \(\frac{3}{2}\) x^3/2 + \(\frac{1}{2}\)x^1/2 + C
हल:

प्रश्न 22.
यदि \(\frac{d}{d x}\) f(x) = 4x³ – \(\frac{3}{x^4}\) जिसमें f(2) = 0 तो f(x)

हल:

⇒ \(\frac{129}{8}\) + C = 0
⇒ C = –\(\frac{129}{8}\)
C का मान समीकरण (1) में रखने पर,
f(x) = x⁴ + \(\frac{1}{x^3}\) – \(\frac{129}{8}\)
अतः विकल्प A सही है।