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Published on: 24/07/2019
Integral Calculus – I
Download Tamil Nadu 12th Standard Business Maths and Statistics question papers, model tests, one-mark questions, important questions, and public exam papers in PDF format. Free study materials and answer keys for TN State Board students.
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1.
Evaluate \(\int { \frac { cos2x-cos2\alpha }{ cosx-cos\alpha } } dx\)
2.
Evaluate \(\int { \frac { { { x }^{ 4 }+{ x }^{ 4 }+1 } }{ { x }^{ 2 }-x+1 } } \)
3.
Integrate the following with respect to x.
\(\sqrt{x}\)(x3 − 2x + 3)
4.
Evaluate \(\int \frac{a x^{2}+b x+c}{\sqrt{x}} d x\)
5.
Evaluate \(\int { \frac { 2{ cos }^{ 2 }x-cos2x }{ { sin }^{ 2 }x } } \)
6.
Evaluate \(\int { { a }^{ 3{ log }_{ a }x } } dx\)
7.
Evaluate \(\int { \frac { { 3x }^{ 2 }+2x+1 }{ x } dx } \)
8.
Integrate the following with respect to x.
\({ \left( { 9x }^{ 2 }-\frac { 4 }{ { x }^{ 2 } } \right) }^{ 2 }\)
9.
Integrate the following with respect to x.
\(\sqrt { 3x+5 } \)
10.
Evaluate \(\int { \frac { dx }{ { \left( 2x+3 \right) }^{ 2 } } } \)
11.
Evaluate \(\int \sqrt{2 x+1} \ d x\)
12.
\(\int { { e }^{ x } } \) (1-cot x +cot2 x) dx = _______________ +c
ex cot x
- ex cot x
ex cosec x
-ex cosec x
13.
\(\int { \frac { 2 }{ { \left( { e }^{ x }+{ e }^{ -x } \right) }^{ 2 } } } \) dx = ____________ +c
\(\frac { { -e }^{ -x } }{ { e }^{ x }+{ e }^{ -x } } \)
\(-\frac { { -e }^{ -x } }{ { e }^{ x }+{ e }^{ -x } } \)
\(\frac { 1 }{ { \left( { e }^{ x }+1 \right) }^{ 2 } } \)
\(\frac { 1 }{ { e }^{ x }-{ e }^{ -x } } \)
14.
\(\int { { \left| x \right| }^{ 3 } } \)dx = ________________ +c
\(\frac { { -x }^{ 4 } }{ 4 } \)
\(\frac { { \left| x \right| }^{ 4 } }{ 4 } \)
\(\frac { { x }^{ 4 } }{ 4 } \)
none of these
15.
If \(\int { \frac { { 2 }^{ \frac { 1 }{ x } } }{ { x }^{ 2 } } } dx=k{ 2 }^{ \frac { 1 }{ x } }\) +c, then k is ___________
\(-\frac { 1 }{ { log }_{ e }2 } \)
- loge2
-1
\(\frac { 1 }{ 2 } \)
16.
\(\int { \left( x-1 \right) } { e }^{ -x }\) dx = __________ +c
-xex
xex
-xe-x
xe-x
17.
\(\int _{ 0 }^{ \infty }{ { x }^{ 4 }{ e }^{ -x } } \)dx is _______.
12
4
4!
64
18.
\(\Gamma \left( \frac { 3 }{ 2 } \right) \) _______.
\(\sqrt { \pi } \)
\(\frac { \sqrt { \pi } }{ 2 } \)
\(2\sqrt { \pi } \)
\(\frac32\)
19.
ഽ\(\frac { sin2x }{ 2sinx } dx\) is _______.
sin x + c
\(\frac12\)sin x + c
cos x + c
\(\frac12\)cos x + c
20.
ഽ2xdx is _______.
2x log 2 + c
2x + c
\(\frac { 2^{ x } }{ log2 } +c\)
\(\frac { log2 }{ { 2 }^{ x } } +c\)
21.
ഽ\(\frac { 1 }{ { x }^{ 3 } } \)dx is _______.
\(\frac { -3 }{ { x }^{ 2 } } +c\)
\(\frac { -1 }{ 2{ x }^{ 2 } } +c\)
\(\frac { -1 }{ { 3x }^{ 2 } } +c\)
\(\frac { -2 }{ { x }^{ 2 } } +c\)
22.
\(\int _{ 0 }^{ \infty }{ { x }^{ n }{ e }^{ -ax } } \) dx where n is a positive integer
23.
For n > 0, \(\int _{ 0 }^{ \infty }{ { x }^{ n-1 }{ e }^{ -x } } \) dx
24.
\(\int _{ 0 }^{ \infty }{ { e }^{ -t } } dt\quad \)
25.
\(\int _{ 0 }^{ 1 }{ { e }^{ -t } } dt\quad \)
26.
∫ e-t dt
27.
If f'(x) = a sin x + b cos x and f'(0) = 4, f(0) = 3, f\(\left( \frac { \pi }{ 2 } \right) \) = 5, find f(x).
1.
\(\int { \frac { cos2x-cos2\alpha }{ cosx-cos\alpha } } dx\)
= \(\frac { ({ 2cos }^{ 2 }x-1)-({ 2cos }^{ 2 }\alpha -1) }{ cos\quad x-cos\alpha } dx\)
= \(\frac { { 2cos }^{ 2 }x-cos^{ 2 }\alpha }{ cos\quad x-cos\alpha } \)
= \(2\int { (cos\quad x+cos\alpha )dx } \)
= \(2[sinx+cos\alpha .x]+c\)
= \(2sinx+2xcos\alpha +c\)
2.
\(\int { \frac { { \left( { x }^{ 2 }+1 \right) }^{ 2 }-{ x }^{ 2 } }{ { x }^{ 2 }-x+1 } } dx=\int { \frac { { \left( { x }^{ 2 }+1 \right) }^{ 2 }-{ x }^{ 2 } }{ { x }^{ 2 }-x+1 } } \)
[∵ a2 - b2 = (a+b) (a-b)]
= ∫ (x2 + 1 + x)dx
= \(\frac { { x }^{ 3 } }{ 3 } +x+\frac { { x }^{ 2 } }{ 2 } +c\)
3.
\(\int { \sqrt { x } } \left( { x }^{ 2 }-2x+3 \right) dx\)
\(=\int { { x }^{ \frac { 1 }{ 2 } } } \left( { x }^{ 3 }-2x+3 \right) dx\)
\(=\int { \left( { x }^{ 3+\frac { 1 }{ 2 } }-{ 2x }^{ 1+\frac { 1 }{ 2 } }+{ 3x }^{ \frac { 1 }{ 2 } } \right) } dx\)
\(=\frac { { x }^{ \frac { 7 }{ 2 } +1 } }{ \frac { 7 }{ 2 } +1 } -\frac { { 2x }^{ \frac { 3 }{ 2 } +1 } }{ \frac { 3 }{ 2 } +1 } +\frac { { 3x }^{ \frac { 1 }{ 2 } +1 } }{ \frac { 1 }{ 2 } +1 } +c\)
\(=\frac { { x }^{ \frac { 9 }{ 2 } } }{ \frac { 9 }{ 2 } } -2\frac { { x }^{ \frac { 5 }{ 2 } } }{ \frac { 5 }{ 2 } } +3\frac { { x }^{ \frac { 3 }{ 2 } } }{ \frac { 3 }{ 2 } } +c\)
\(=\frac { 2 }{ 9 } { x }^{ \frac { 9 }{ 2 } }-\frac { 4 }{ 5 } { x }^{ \frac { 5 }{ 2 } }+{ 2x }^{ \frac { 3 }{ 2 } }+c\)
4.
\(\int { \frac { { ax }^{ 2 }+bx+c }{ \sqrt { x } } dx } \)
=\(\int { \left( { ax }^{ \frac { 2 }{ 3 } }+{ bx }^{ \frac { 2 }{ 3 } }+{ cx }^{ -\frac { 1 }{ 2 } } \right) dx } \)
=\(a\int { { x }^{ \frac { 3 }{ 2 } }dx+b\int { { x }^{ \frac { 1 }{ 2 } }+c\int { { x }^{ -\frac { 1 }{ 2 } }dx } } } \)
=\(\frac { { 2ax }^{ \frac { 5 }{ 2 } } }{ 5 } +\frac { { 2bx }^{ \frac { 3 }{ 2 } } }{ 3 } +{ 2cx }^{ \frac { 1 }{ 2 } }+k\)
5.
∴ \(\int { \frac { 2{ cos }^{ 2 }x-cos2x }{ { sin }^{ 2 }x } } \)
= \(\int { \frac { 2{ cos }^{ 2 }x-\left( 2{ cos }^{ 2 }x-1 \right) }{ { sin }^{ 2 }x } } \)
= \(\int { \frac { 1 }{ { sin }^{ 2 }x } } dx\)
= ∫ cosec2 x dx
= -cot x + c
6.
We know that logax = x
∴ ∫ a3logax dx = ∫ alogax3 dx = ∫ x3 dx
= \(\frac { { x }^{ 4 } }{ 4 } +c\)
7.
\(\int { \frac { { 3x }^{ 2 }+2x+1 }{ x } dx } =\int { \left( 3x+2+\frac { 1 }{ x } \right) dx } \)
\(=\frac { { 3x }^{ 2 } }{ 2 } +2x+log\left| x \right| +c\)
8.
\(\int { { \left( { 9x }^{ 2 }-\frac { 4 }{ { x }^{ 2 } } \right) }^{ 2 } } dx\)
\(\left[ \because { \left( a-b \right) }^{ 2 }={ a }^{ 2 }-2ab+{ b }^{ 2 } \right] \)
\(=\int { \left( { 81x }^{ 4 }-72+\frac { 16 }{ { x }^{ 4 } } \right) } dx\)
\(=81\frac { { x }^{ 4+1 } }{ 4+1 } -72x+16\frac { { x }^{ -4+1 } }{ -4+1 } +c\)
\(\left[ \because \frac { 16 }{ { x }^{ 4 } } =16{ x }^{ -4 } \right] \)
\(=81\frac { { x }^{ 5 } }{ 5 } -72x+16\frac { { x }^{ -3 } }{ -3 } +c\)
\(=\frac { 81 }{ 5 } { x }^{ 5 }-72x-\frac { 16 }{ { 3x }^{ 3 } } +c\)
9.
\(\int { \sqrt { 3x+5 } dx } \)
\(=\int { { \left( 3x+5 \right) }^{ 1/2 } } dx\)
\(=\frac { { \left( 3x+5 \right) }^{ 1/2+1 } }{ 3\left( \frac { 1 }{ 2 } +1 \right) } +c\)
\(\left[ \because \int { { \left( ax+b \right) }^{ n }dx=\frac { { \left( ax+b \right) }^{ n+1 } }{ a\left( n+1 \right) } } +c \right] \)
\(=\frac { { \left( 3x+5 \right) }^{ 3/2 } }{ 3\left( \frac { 3 }{ 2 } \right) } +c=\frac { { \left( 3x+5 \right) }^{ 3/2 } }{ \frac { 9 }{ 2 } } +c\)
\(=\frac { 2 }{ 9 } { \left( 3x+5 \right) }^{ 3/2 }+c\)
10.
\(\int { \frac { dx }{ { \left( 2x+3 \right) }^{ 2 } } } \) =\(\int { { \left( 2x+3 \right) }^{ -2 }dx } \)
\(=-\frac { 1 }{ 2\left( 2x+3 \right) } +c\)
11.
\( \int \sqrt{2 x+1} \ d x=\int(2 x+1)^{\frac{1}{2}} d x\)
\(=\frac { { \left( 2x+1 \right) }^{ \frac { 3 }{ 2 } } }{ 3 } +c\)
12.
(b)
- ex cot x
13.
(a)
\(\frac { { -e }^{ -x } }{ { e }^{ x }+{ e }^{ -x } } \)
14.
(d)
none of these
15.
(a)
\(-\frac { 1 }{ { log }_{ e }2 } \)
16.
(c)
-xe-x
17.
(c)
4!
18.
(b)
\(\frac { \sqrt { \pi } }{ 2 } \)
19.
(a)
sin x + c
20.
(c)
\(\frac { 2^{ x } }{ log2 } +c\)
21.
(b)
\(\frac { -1 }{ 2{ x }^{ 2 } } +c\)
22.
\(\frac { n! }{ { a }^{ n+1 } } \)
23.
\(\Gamma (n)\)
24.
Improper definite intgral
25.
proper definite integer
26.
Indefinite inegral
27.
Given f'(x) = a sin x + b cos x ------(1)
ഽ f'(x) dx = a ഽ sin x dx + b ഽ cos x dx [∵ f'(x) dx = f(x)]
⇒ f(x) = -a cos x + b sin x + c -------(2)
Given f'(0) = 4
∴ (1) 4 = a sin 0 + b cos 0
⇒ 4 = a (0) + b (1)
[∵ sin 0 = 0 and cos 0 = 1]
⇒ 4 = b
Also, f(0) = 3
∴ (2) 3 = -a cos 0 + b sin 0 +c
⇒ 3 = -a (1) + b (0) +c
⇒ 3 = -a +c --------(3)
And f\(\left( \frac { \pi }{ 2 } \right) \) = 5
∴ (2) 5 = -a cos \( \frac { \pi }{ 2 } \)+ b sin \( \frac { \pi }{ 2 } \)+ c
⇒ 5 = -a (0) + b (1) +c
[∵ sin \( \frac { \pi }{ 2 } \)=1 and cos \( \frac { \pi }{ 2 } \)= 0]
⇒ 5 = b + c
⇒ 5 = 4 + c [∵ b = 4]
⇒ 5 - 4 = c
c = 1
Substituting c = 1 in (3) we get,
3 = -a + 1
⇒ a = 1 - 3
⇒ a = -2
∴ From (2), f(x) = -(-2) cos x + 4 sin x + 1
⇒ f(x) = 2 cos x + 4 sin x +1
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