These NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1 Questions and Answers are prepared by our highly skilled subject experts. https://mcq-questions.com/ncert-solutions-for-class-12-maths-chapter-7-ex-7-1/
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.1
Question 1.
sin 2x
Solution:
Question 2.
cos 3x
Solution:
Question 3.
\({ e }^{ 2x }\)
Solution:
Question 4.
(ax + c)²
Solution:
Question 5.
sin 2x – 4 e3x
Solution:
∴ An anti derivative of sin2x – 4e3x is – \(\frac { 1 }{ 2 }\) cos 2x – \(\frac { 4 }{ 3 }\)4e3x
Question 6.
\(\int { \left( { 4e }^{ 3x }+1 \right) dx } \)
Solution:
Question 7.
\(\int { { x }^{ 2 }\left( 1-\frac { 1 }{ { x }^{ 2 } } \right) dx } \)
Solution:
\(=\int { { x }^{ 2 }\left( 1-\frac { 1 }{ { x }^{ 2 } } \right) }dx
= ∫(x² – 1)dx
= ∫x² dx – ∫1 dx
= [latex]\frac { { x }^{ 3 } }{ 3 }\) – x + C
Question 8.
\(\int { { (ax }^{ 2 }+bx+c)dx } \)
Solution:
\(\int { { (ax }^{ 2 }+bx+c)dx } \)
= a∫x² dx + b ∫x dx + c∫1 dx
= \(\frac { { ax }^{ 3 } }{ 3 } +\frac { { bx }^{ 2 } }{ 2 }\) + cx + C
Question 9.
\(\int { \left( { 2x }^{ 2 }+{ e }^{ x } \right) dx } \)
Solution:
\(\int { \left( { 2x }^{ 2 }+{ e }^{ x } \right) dx } \)
= 2∫x² dx + ∫ex dx
= 2 \(\frac { { x }^{ 3 } }{ 3 }\) + ex + C
= \(\frac { { 2x }^{ 3 } }{ 3 } +{ e }^{ x }\) + C
Question 10.
\(\int { { \left[ \sqrt { x } -\frac { 1 }{ \sqrt { x } } \right] }^{ 2 }dx } \)
Solution:
Question 11.
\(\int { \frac { { x }^{ 3 }+{ 5x }^{ 2 }-4 }{ { x }^{ 2 } } dx } \)
Solution:
Question 12.
\(\int { \frac { { x }^{ 3 }+3x+4 }{ \sqrt { x } } dx } \)
Solution:
Question 13.
\(\int { \frac { { x }^{ 3 }-{ x }^{ 2 }+x-1 }{ x-1 } dx } \)
Solution:
\(=\int { \frac { { x }^{ 2 }(x-1)+(x-1) }{ x-1 } dx } \)
\(=\int { \left( { x }^{ 2 }+1 \right) dx } =\frac { { x }^{ 3 } }{ 3 } +x+c \)
Question 14.
\(\int { \left( 1-x \right) \sqrt { x } dx } \)
Solution:
Question 15.
\(\int { \sqrt { x } \left( { 3x }^{ 2 }+2x+3 \right) dx } \)
Solution:
Question 16.
\(\int { (2x – 3cosx+{ e }^{ x })dx } \)
Solution:
\(\int { (2x – 3cosx+{ e }^{ x })dx } \)
= 2∫x dx – 3∫cosx dx + ∫ex dx
= 2(\(\frac { { x }^{ 2 } }{ 3 }\)) – 3 sin x + ex + C
\(={ x }^{ 2 }-3sinx+{ e }^{ x }\) + C
Question 17.
\(\int { \left( { 2x }^{ 2 }-3sinx+5\sqrt { x } \right) dx } \)
Solution:
Question 18.
\(\int { secx(secx+tanx)dx } \)
Solution:
\(\int { secx(secx+tanx)dx } \)
= \(\int\left(\sec ^{2} x+\sec x \tan x\right) d x\)
= ∫sec² x dx + ∫sec x tan x dx
= tan x + sec + C
Question 19.
\(\int { \frac { { sec }^{ 2 }x }{ { cosec }^{ 2 }x } dx } \)
Solution:
= \(\int { \frac { 1 }{ { cos }^{ 2 }x } } { sin }^{ 2 }xdx\)
= \(\int { tan } ^{ 2 }xdx\quad\)
= \(\int\left(\sec ^{2} x-1\right) d x\)
= tanx – x + c
Question 20.
\(\int { \frac { 2-3sinx }{ { cos }^{ 2 }x } dx } \)
Solution:
= \(\int { \left( \frac { 2 }{ { cos }^{ 2 }x } -3\frac { sinx }{ { cos }^{ 2 }x } \right) dx } \)
= \(\int { ({ 2sec }^{ 2 }x-3secxtanx)dx }\)
= 2tanx – 3secx + c
Choose the correct answer in Exercises 21 and 22.
Question 21.
The anti derivative \(\left( \sqrt { x } +\frac { 1 }{ \sqrt { x } } \right) \) equals
(a) \(\frac { 1 }{ 3 } { x }^{ \frac { 1 }{ 3 } }+{ 2x }^{ \frac { 1 }{ 2 } }+c\)
(b) \(\frac { 2 }{ 3 } { x }^{ \frac { 2 }{ 3 } }+{ \frac { 1 }{ 2 } x }^{ 2 }+c\)
(c) \(\frac { 2 }{ 3 } { x }^{ \frac { 3 }{ 2 } }+{ 2x }^{ \frac { 1 }{ 2 } }+c\)
(d) \(\frac { 3 }{ 2 } { x }^{ \frac { 3 }{ 2 } }+\frac { 1 }{ 2 } { x }^{ \frac { 1 }{ 2 } }+c\)
Solution:
(c) \(\frac { 2 }{ 3 } { x }^{ \frac { 3 }{ 2 } }+{ 2x }^{ \frac { 1 }{ 2 } }+c\)
Question 22.
If \(\frac { d }{ dx } f(x) = { 4x }^{ 3 } -\frac { 3 }{ { x }^{ 4 } } \) such that f(2) = 0 then f(x) is
(a) \({ x }^{ 4 }+\frac { 1 }{ { x }^{ 3 } } -\frac { 129 }{ 8 } \)
(b) \({ x }^{ 3 }+\frac { 1 }{ { x }^{ 4 } } +\frac { 129 }{ 8 } \)
(c) \({ x }^{ 4 }+\frac { 1 }{ { x }^{ 3 } } +\frac { 129 }{ 8 } \)
(d) \({ x }^{ 3 }+\frac { 1 }{ { x }^{ 4 } } -\frac { 129 }{ 8 } \)
Solution:
(a) \({ x }^{ 4 }+\frac { 1 }{ { x }^{ 3 } } -\frac { 129 }{ 8 } \)
