2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals

Students can Download Basic Maths Question Bank Chapter 20 Indefinite Integrals Questions and Answers, Notes Pdf, 2nd PUC Basic Maths Question Bank with Answers helps you to revise the complete Karnataka State Board Syllabus and to clear all their doubts, score well in final exams.

Karnataka 2nd PUC Basic Maths Question Bank Chapter 20 Indefinite Integrals

2nd PUC Basic Maths Indefinite Integrals One Mark Questions and Answers

Question 1.
∫(7x – 3)⁴ dx
Answer:
\(=\frac{(7 x-3)^{5}}{35}+\mathrm{C}\)

Question 2.
∫(2x + 5)^3/5 dx
Answer:

Question 3.
\(\int \frac{1}{10 x+3} d x\)
Answer:
\(=\frac{\log (10 x+3)}{10}+C\)

Question 4.
∫e^{3 – 4x} dx
Answer:
\(\frac{e^{3-4 x}}{-4}+C\)

Question 5.
∫3^{5x – 3}
Answer:
\(\frac{3^{5 x-3}}{5 . \log 3}+\mathrm{C}\)

Question 6.
∫sec²(x – 5) dx
Answer:
= tan(x – 5) + C

Question 7.
∫cosec(3 – 5x).cot(3 – 5x) dx
Answer:

Question 8.
\(\int \frac{5}{\sec x} d x\)
Answer:
sec x
= 5∫cos x = 5. sin x + C

Question 9.
\(\int \frac{8}{\sin ^{2} x} d x\)
Answer:
= ∫8.cosec² x dx ,
= -8 cot x + C

Question 10.
∫tan² x dx
Answer:
∫(sec² x – 1)dx = tan x – x + C

Question 11.
Integrate \(\frac{x^{2}-1}{x}\) w.r.t x
Answer:

Question 12.
Integrate \(\frac{1}{3-4 x}\) w.r.t x
Answer:

Question 13.
Integrate \(\frac{x+5}{x+2}\) w.r.t x
Answer:

= x + 3 log (x + 2) + C

Question 14.
∫x. √xdx = ∫x^3/2 dx
Answer:

Question 15.
\(\int \frac{x+1}{x} d x\)
Answer:
= ∫1 + \(\frac{1}{x}\) dx = x + log x + C

Question 16.
\(\int \sqrt{a x+b} d x\)
Answer:

Question 17.

Answer:
= log (e^x + 1) + C

Question 18.
\(\int \frac{1}{2 \sqrt{x}} d x\)
Answer:
= √x + C

Question 19.
Evaluate \(\int \frac{x}{1+x^{2}} d x\)
Answer:

Question 20.
Evaluate \(\int \frac{1}{x}-\frac{1}{x^{2}} d x\)
Answer:
= log x + \(\frac{1}{x}\) + C

Question 21.
Evaluate \(\int \frac{x}{x+4} d x\)
Answer:

= x – 4 log(x + C) + C

Question 22.

Answer:

Question 23.
Evaluate ∫(X^e + e^x + e^e) dx
Answer:
= \(\frac{x^{e+1}}{e+1}\) + e^x + e^ex + C

Question 24.
Evaluate \(\int\left(x+\frac{1}{x}\right) d x\)
Answer:
= \(\frac{x^{2}}{2}\) + log x + C

Question 25.
Evaluate \(\int \frac{1}{e^{x}} d x\)
Answer:
= ∫e^-x dx = -e^-x + C

Question 26.
Evaluate \(\int(\sqrt{x}+x+1) d x\)
Answer:
\(=\frac{2}{3} x^{\frac{3}{2}}+\frac{x^{2}}{2}+x+C\)

Question 27.
Evaluate \(\int \sqrt{3 x-5} d x\)
Answer:
\(=\frac{2(3 x-5)^{\frac{3}{2}}}{9}+C\)

Question 28.
Evaluate ∫e^-5x dx
Answer:
\(=\frac{e^{-5 x}}{-5}+C\)

Question 29.
\(\int \frac{1}{x \log x} d x\)
Answer:

Question 30.
Evaluate \(\int \frac{1}{(4 x+1)^{3}} d x\)
Answer:

Question 31.
Evaluate \(\int \frac{\sin x-\cos x}{\sin x+\cos x} d x\)
Answer:
log (sin x + cos x) + C

Question 32.
Evaluate ∫(x² + 2x – 5)⁴ (x + 1)dx
Answer:

2nd PUC Basic Maths Indefinite Integrals Two or Three Marks Questions and Answers

Question 1.
∫sin 3x . cos x dx
Answer:
By transformation formula
= ∫\(\frac { 1 }{ 2 }\)(sin 4x + sin 2x)dx
= \(\frac{1}{2}\left[\frac{-\cos 4 x}{4}-\frac{\cos 2 x}{2}\right]+C\)

Question 2.
∫sin 4x . sin 2x dx
Answer:
= ∫\(\frac { 1 }{ 2 }\)[cos 2x – cos 6x] dx
\(\frac{1}{2}\left[\frac{\sin 2 x}{2}-\frac{\sin 6 x}{6}\right]+C\)

Question 3.
∫cos 3x . cos 2x dx
Answer:
= ∫\(\frac { 1 }{ 2 }\)[cos(5x) + cos x] dx
= \(\frac{1}{2}\left[\frac{\sin 5 x}{5}+\sin x\right]+C\)

Question 4.
∫cos 5x . sin x dx
Answer:
∫\(\frac { 1 }{ 2 }\) [sin 6x – sin 4x]dx
\(\frac{1}{2}\left[-\frac{\cos 6 x}{6}+\frac{\cos 4 x}{4}\right]+C\)

Question 5.
∫cos² 3x dx
Answer:

Question 6.
∫sin² xdx
Answer:

Question 7.
∫sin x . cos x dx
Answer:

Question 8.
∫sin² 2x . cos² 2x dx
Answer:

Question 9.
∫sin³ x dx
Answer:

Question 10.
∫ cos² x dx
Answer:

Question 11.
\(\int \frac{\sin ^{2} x}{1+\cos x}\)
Answer:
= \(\int \frac{1-\cos ^{2} x}{1+\cos ^{2} x} d x\)
= ∫1 + cos x dx = x + sin x + C

Question 12.
\(\int \frac{1}{1+\cos x} d x\)
Answer:

Question 13.
\(\int \frac{1}{1-\cos 2 x} d x\)
Answer:
\(=\int \frac{1}{2 \sin ^{2} x} d x\)
= \(\frac { 1 }{ 2 }\) ∫cosec² x dx
= \(\frac { 1 }{ 2 }\) (-cot x) + C

Question 14.

Answer:
= ∫cosec x\(\sqrt{\cot ^{2} x} d x\)
= ∫cosec x cot x dx = -cosecx+ c

Question 15.
\(\int \sqrt{1+\sin 2 x} d x\)
Answer:

= -cos x + sin x + c.

Question 16.
\(\int \frac{1-\cos 2 x}{1+\cos 2 x} d x\)
Answer:

= ∫(sec²x – 1)dx
= tanx – x + c.

Question 17.
\(\int \frac{1}{1+\sin x} d x\)
Answer:

= ∫sec²x – ∫tan x sec x
= tan x – sec x + c

Question 18.
\(\int \frac{1+\sin x}{\cos ^{2} x} d x\)
Answer:

= ∫sec² x + sec x . tan x dx
= tan x – sec x + cos

Question 19.

Answer:

Question 20.
\(\int \frac{4}{\sqrt{x+1}+\sqrt{x+2}} d x\)
Answer:

Question 21.
\(\int \frac{7 x^{6}+7^{x} \log 7}{x^{7}+7^{x}} d x\)
Answer:
= log (x⁷ + 7^x) + C

Question 22.
\(\int \frac{3^{x} \log 3}{3^{x}+1} d x\)
Answer:
= log (3^x + 1)

Question 23.
\(\int \sqrt{x}\left(1-\frac{1}{x}\right) d x\)
Answer:

Question 24.
\(\int \frac{d x}{1+e^{-x}}\)
Answer:

= log (e^x + 1) + C

Question 25.

Answer:

Question 26.
\(\int \frac{d x}{1+e^{x}} d x\)
Answer:

Question 27.
\(\int \frac{1}{1+2 e^{x}} d x\)
Answer:

= – log(e^-x + 2) + C

Question 28.
∫xe^5x dx
Answer:

Question 29.

Answer:

Question 30.
\(\int \frac{\sqrt{4+\log x}}{x} d x\)
Answer:

Question 31.
\(\int \frac{2 x^{2}+5 x+1}{x} d x\)
Answer:

Question 32.
∫e^x3 . 3x² dx
Answer:
∫e^t . dt
e^t + c
= e^x3 + c
put x³ = t
3x² dx = dt

Question 33.
∫x log x dx
Answer:
∫log x . dx

2nd PUC Basic Maths Indefinite Integrals Four or Five Marks Questions and Answers

Question 1.
\(\int \frac{1}{\sqrt{x+1}-\sqrt{x-2}} d x\)
Answer:

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

3x – 1 = A(x – 2)(x – 3) + B(x – 1)(x – 3) + C(x – 1)(x – 2)
put x = 1
3 – 1 = A(1 – 2)(1 – 3) + B(0) + C(0)
2 = A(-1) (-2) ⇒ A = \(\frac{2}{2}\) = 1
put x = 2
6 – 1 = A(0) + B(2 – 1)(2 – 3) + C(0)
5 = B(1)(-1) ⇒B = -5
put x = 3
9 – 1 = A(0) + B(0) + C(3 – 1)(3 – 2)
8 = C(2)(1) ⇒ C = \(\frac{8}{2}\) = 4

= log (x – 1) -5 log (x – 2) + 4 log (x – 3) + C

Question 3.
\(\int \frac{(\log x)^{n}}{x} d x\)
Answer:

Question 4.
\(\int \frac{1}{a^{2}-x^{2}} d x\)
Answer:

Finding A & B using parital fractions we get

Question 5.
∫(2x² – 6x + 4)^3/2 (2x – 3) dx
Answer:
put 2x² – 6x + 4 = t
(4x – 6) dx = dt
2(2x – 3) dx = dt
(2x – 3) dx = \(\frac{d t}{2}\)

Question 6.

Answer:

Question 7.

Answer:


-x²e^-x – 2xe^-x + ∫e^-x dx
= -x²e^-x – 2xe^-x – e^-x + C

Question 8.
\(\int \frac{x^{2}}{(x-1)(x-2)(x-3)} d x\)
Answer:

By using partial fractions find the values A, B & C we get
A = \(\frac { 1 }{ 2 }\) , B = -4 , C = \(\frac { 9 }{ 2 }\)

Question 9.
\(\int \frac{1}{x(x-2)(x+3)} d x\)
Answer:
By using partial fractions find the values of A, B & C we get

Question 10.
\(\int \frac{x^{3}}{x^{2}+7 x+12} d x\)
Answer:
This is an improper fraction

Question 11.
\(\int \frac{3 x+2}{(x-1)(2 x+3)} d x\)
Answer:
= \(\int \frac{A}{x-1}+\frac{B}{2 x+3} d x\)
Resolving by partial fractions we get
A = 1, B = 1

Question 12.
\(\int \frac{2 x+3}{(x+1)^{2}(x-3)} d x\)
Answer:

By Resolving partial fractions
we get A = \(\frac { -9 }{ 16 }\), B = \(\frac { -1 }{ 4 }\), C = \(\frac { 9 }{ 16 }\)

Question 13.
\(\int \frac{2 x}{\left(x^{2}+4 x+4\right)(x-1)} d x\)
Answer:

By Resolving in to partial fractions find the values of A, B & C we get
A = \(\frac { -2 }{ 9 }\), B = \(\frac { 4 }{ 3 }\), C = \(\frac { 2 }{ 9 }\)

Question 14.

Answer:

Resolving partial fractions find the values of A &B we get A = -1, B = 1
Put log x = t
\(\frac{1}{x}\) dx = dt

Question 15.
\(\int \frac{e^{x}}{4 e^{2 x}-4 e^{x}-3} d x\)
Answer:
put e^x = t ∴e^x dx = dt

Find the values of A & B using partial fractions we get A = -1/4 , B = 1/4