NCERT Solutions for Class 12 Mathemetics Chapter 7 - Integrals

Question 2:

Find an anti-derivative (or integral) of the following by the method of inspection :
sin 2x

Question 3:

Find an anti-derivative (or integral) of the following by the method of inspection :
cos 3x

Question 4:

Find an anti-derivative (or integral) of the following by the method of inspection :
e^2x

Question 5:

Find an anti-derivative (or integral) of the following by the method of inspection :
(ax + b)3

Question 6:

Find an anti-derivative (or integral) of the following by the method of inspection :
sin 2x – 4e^3x

Question 7:

Find the following integrals in

Question 8:

Find the following integrals in

Question 9:

Find the following integrals in

Question 10:

Find the following integrals in

Question 11:

Find the following integrals in

Question 12:

Find the following integrals in

Question 13:

Find the following integrals in

Question 14:

Find the following integrals in

Question 15:

Find the following integrals in

Question 16:

Find the following integrals in

Question 17:

Find the following integrals in

Question 18:

Find the following integrals in

Question 19:

Find the following integrals in

Question 20:

Find the following integrals in

Question 21:

Find the following integrals in

Question 22:

Question 23:

Question 24:

Integrate the following

Question 25:

Integrate the following

Question 26:

Integrate the following

Question 27:

Integrate the following
sin x sin (cos x)

Question 28:

Integrate the following
sin (ax + b) cos (ax + b)

Question 29:

Integrate the following

Question 30:

Integrate the following

Question 31:

Integrate the following

Question 32:

Integrate the following

Question 33:

Integrate the following

Question 34:

Integrate the following

Question 35:

Integrate the following

Question 36:

Integrate the following

Question 38:

Integrate the following

Question 39:

Integrate the following

Question 40:

Integrate the following
e^{2x + 3}

Question 41:

Integrate the following

Question 42:

Integrate the following

Question 43:

Integrate the following

Question 44:

Integrate the following

Question 45:

Integrate the following
tan² (2x – 3)

Question 46:

Integrate the following
sec² (7 – 4x).

Question 47:

Integrate the following

Question 48:

Integrate the following

Question 49:

Integrate the following

Question 50:

Integrate the following

Question 51:

Integrate the following

Question 52:

Integrate the following

Question 53:

Integrate the following
cot x log sin x

Question 54:

Integrate the following

Question 55:

Integrate the following

Question 56:

Integrate the following

Question 57:

Integrate the following

Question 58:

Integrate the following

Question 59:

Integrate the following

Question 60:

Integrate the following

Question 61:

Integrate the following

Question 62:

Choose the correct answer

1. 10^x – x¹⁰ + C
2. 10^x + x¹⁰ + C
3. (10^x – x¹⁰)^{– 1} + C
4. log (10^x + x¹⁰) + C

Question 63:

Choose the correct answer

1. tan x + cot x + C
2. tan x – cot x + C
3. tan x cot x + C
4. tan – cot 2x + C

Question 64:

Find the integrals of the functions in
sin² (2x + 5)

Question 65:

Find the integrals of the functions in
sin 3x cos 4x

Question 66:

Find the integrals of the functions in
cos 2x cos 4x cos 6x

Question 67:

Find the integrals of the functions in
sin³ (2x + 1)

Question 68:

Find the integrals of the functions in
sin³ x cos³ x.

Question 69:

Find the integrals of the functions in
sin x sin 2x sin 3x

Question 70:

Find the integrals of the functions in
sin 4x sin 8x

Question 71:

Find the integrals of the functions in

Question 72:

Find the integrals of the functions in

Question 73:

Find the integrals of the functions in
sin⁴ x.

Question 74:

Find the integrals of the functions in
cos⁴ 2x.

Question 75:

Find the integrals of the functions in

Question 76:

Find the integrals of the functions in

Question 77:

Find the integrals of the functions in

Question 78:

Find the integrals of the functions in
tan³ 2x sec 2x

Question 79:

Find the integrals of the functions in
tan⁴x

Question 80:

Find the integrals of the functions in

Question 81:

Find the integrals of the functions in

Question 82:

Find the integrals of the functions in

Question 83:

Find the integrals of the functions in

Question 84:

Find the integrals of the functions in
sin^–1 (cos x).

Question 85:

Find the integrals of the functions in

Question 86:

1. tan x + cot x + C
2. tan x + cosec x + C
3. tan (e^x) + C
4. cot (e^x) + C

Question 87:

1. – cot (e^xx)
2. tan (x^ex) + C
3. tan (e^x) + C
4. cot (e^x) + C

Question 88:

Integrate the functions in

Question 89:

Integrate the functions in

Question 90:

Integrate the functions in

Question 91:

Integrate the functions in

Question 92:

Integrate the functions in

Question 93:

Integrate the functions in

Question 94:

Integrate the functions in

Question 95:

Integrate the functions in

Question 96:

Integrate the functions in

Question 97:

Integrate the functions in

Question 98:

Integrate the functions in

Question 99:

Integrate the functions in

Question 100:

Integrate the functions in

Question 101:

Integrate the functions in

Question 102:

Integrate the functions in

Question 103:

Integrate the functions in

Question 104:

Integrate the functions in

Question 105:

Integrate the functions in

Question 106:

Integrate the functions in

Question 107:

Integrate the functions in

Question 108:

Integrate the functions in

Question 109:

Integrate the functions in

Question 110:

Integrate the functions in

Question 111:

Choose the correct answer in

1. x tan^–1 (x + 1) + C
2. tan^–1 (x + 1) + C
3. (x + 1) tan^–1x + C
4. tan^–1 x + C

Question 112:

Choose the correct answer in

Question 113:

Integrate the rational functions in

Question 114:

Integrate the rational functions in

Question 115:

Integrate the rational functions in

Question 116:

Integrate the rational functions in

Question 117:

Integrate the rational functions in

Question 118:

Integrate the rational functions in

Question 119:

Integrate the rational functions in

Question 120:

Integrate the rational functions in

Question 121:

Integrate the rational functions in

Question 122:

Integrate the rational functions in

Question 123:

Integrate the rational functions in

Question 124:

Integrate the rational functions in

Question 125:

Integrate the rational functions in

Question 126:

Integrate the rational functions in

Question 127:

Integrate the rational functions in

Question 128:

Integrate the rational functions in

Question 129:

Integrate the rational functions in

Question 130:

Integrate the rational functions in

Question 131:

Integrate the rational functions in

Question 132:

Integrate the rational functions in

Question 133:

Integrate the rational functions in

Question 134:

Choose the correct answer

1. 

2. 

3. 

4. log | (x – 1) (x – 2) | + C

Question 135:

Choose the correct answer

Question 136:

Integrate the functions in
x sin x

Question 137:

Integrate the functions in
x sin 3x

Question 138:

Integrate the functions in
x²e^x

Question 139:

Integrate the functions in
x log x

Question 140:

Integrate the functions in
x log 2x

Question 141:

Integrate the functions in
x² log x

Question 142:

Integrate the functions in
x sin^–1 x

Question 143:

Integrate the functions in
x tan^–1 x

Question 144:

Integrate the functions in
x cos^–1 x

Question 145:

Integrate the functions in
(sin^–1 x)²

Question 146:

Integrate the functions in

Question 147:

Integrate the functions in
x sec² x.

Question 148:

Integrate the functions in
tan^–1 x.

Question 149:

Integrate the functions in
x (log x)².

Question 150:

Integrate the functions in
(x² + 1) log x.

Question 151:

Integrate the functions in
e^x(sin x + cos x).

Question 152:

Integrate the functions in

Question 153:

Integrate the functions in

Question 154:

Integrate the functions in

Question 155:

Integrate the functions in

Question 156:

Integrate the functions in
e^2x sin x.

Question 157:

Integrate the functions in

Question 158:

Choose the correct answer in

Question 159:

Choose the correct answer in

1. e^x cos x +C
2. e^x sec x + C
3. e^x sin x + C
4. e^x tan x + C

Question 160:

Integrate the functions in

Question 161:

Integrate the functions in

Question 162:

Integrate the functions in

Question 163:

Integrate the functions in

Question 164:

Integrate the functions in

Question 165:

Integrate the functions in

Question 166:

Integrate the functions in

Question 167:

Integrate the functions in

Question 168:

Integrate the functions in

Question 169:

Choose the correct answer

Question 170:

Choose the correct answer

Question 171:

Evaluate the following definite integrals as limit of sums :

Question 172:

Evaluate the following definite integrals as limit of sums :

Question 173:

Evaluate the following definite integrals as limit of sums :

Question 174:

Evaluate the following definite integrals as limit of sums :

Question 175:

Evaluate the following definite integrals as limit of sums :

Question 176:

Evaluate the following definite integrals as limit of sums :

Question 178:

Evaluate the definite integrals

Question 179:

Evaluate the definite integrals

Question 180:

Evaluate the definite integrals

Question 181:

Evaluate the definite integrals

Question 182:

Evaluate the definite integrals

Question 183:

Evaluate the definite integrals

Question 184:

Evaluate the definite integrals

Question 185:

Evaluate the definite integrals

Question 186:

Evaluate the definite integrals

Question 187:

Evaluate the definite integrals

Question 188:

Evaluate the definite integrals

Question 189:

Evaluate the definite integrals

Question 190:

Evaluate the definite integrals

Question 191:

Evaluate the definite integrals

Question 192:

Evaluate the definite integrals

Question 193:

Evaluate the definite integrals

Question 194:

Evaluate the definite integrals

Question 195:

Evaluate the definite integrals

Question 196:

Evaluate the definite integrals

Question 197:

Evaluate the definite integrals

Question 198:

Choose the correct answer

Question 199:

Choose the correct answer

Question 200:

Evaluate the integrals, using substitution

Question 201:

Evaluate the integrals, using substitution

Question 202:

Evaluate the integrals, using substitution

Question 203:

Evaluate the integrals, using substitution

Question 204:

Evaluate the integrals, using substitution

Question 205:

Evaluate the integrals, using substitution

Question 206:

Evaluate the integrals, using substitution

Question 207:

Evaluate the integrals, using substitution

Question 208:

Choose the correct answer

1. 6
2. 0
3. 3
4. 4

Question 209:

Choose the correct answer

1. cos x + x sin x
2. x sin x
3. x cos x
4. sin x + x cos x

Question 210:

By using the properties of definite integrals, evaluate the integrals in

Question 211:

By using the properties of definite integrals, evaluate the integrals in

Question 212:

By using the properties of definite integrals, evaluate the integrals in

Question 213:

By using the properties of definite integrals, evaluate the integrals in

Question 214:

By using the properties of definite integrals, evaluate the integrals in

Question 215:

By using the properties of definite integrals, evaluate the integrals in

Question 216:

By using the properties of definite integrals, evaluate the integrals in

Question 217:

By using the properties of definite integrals, evaluate the integrals in

Question 218:

By using the properties of definite integrals, evaluate the integrals in

Question 219:

By using the properties of definite integrals, evaluate the integrals in

Question 220:

By using the properties of definite integrals, evaluate the integrals in

Question 221:

By using the properties of definite integrals, evaluate the integrals in

Question 222:

By using the properties of definite integrals, evaluate the integrals in

Question 223:

By using the properties of definite integrals, evaluate the integrals in

Question 224:

By using the properties of definite integrals, evaluate the integrals in

Question 225:

By using the properties of definite integrals, evaluate the integrals in

Question 226:

By using the properties of definite integrals, evaluate the integrals in

Question 227:

By using the properties of definite integrals, evaluate the integrals in

Question 228:

Show that

Question 229:

The value of

1. 0
2. 2
3. $\mathrm{\pi }$
4. 1

Question 230:

1. 2

2. 

3. 0
4. -1

Question 231:

Question 232:

Question 233:

1. 10^x – x¹⁰ + c
2. 10^x + x¹⁰ + c
3. (10^x – x¹⁰)^–1 + c
4. log (10^x + x¹⁰) + c.

Question 234:

1. tan x + cot x + c
2. tan x – cot x + c
3. tan x cot x + c
4. tan x – cot 2x + c

Question 235:

1. tan x + cot x + c
2. tan x + cosec x + c
3. – tan x + cot x + c
4. tan x + sec x + c.

Question 236:

1. – cot (xe^x) + c
2. tan (xe^x) + c
3. tan (e^x) + c
4. cot (e^x) + c.

Question 237:

1. x tan^–1 (x + 1) + c
2. tan^–1 (x + 1) + c
3. (x + 1) tan^–1 x + c
4. tan^–1 x + c

Question 238:

Question 239:

1. 

2. 

3. 

4. log | (x – 1) (x – 2) | + c.

Question 240:

1. 

2. 

3. 

4. log | x | + log (x² + 1) + c.

Question 241:

Question 242:

1. e^x cos x + c
2. e^x sec x + c
3. e^x sin x + c
4. e^x tan x + c.

Question 243:

Question 244:

Question 245:

Question 246:

Question 247:

1. 6
2. 0
3. 3
4. 4

Question 248:

1. cos x + x sin x
2. x sin x
3. x cos x
4. sin x + x cos x.

Question 249:

1. 0
2. 2
3. $\mathrm{\pi }$
4. 1

Question 250:

1. 2

2. 

3. 0
4. -2

Question 251:

1. tan^–1 (ex) + c
2. tan^–1 (e– x) + c
3. log (e^x – e^–x) + c
4. log (e^x + e^–x) + c.

Question 252:

1. 

2. log | sin x + cos x | + c.
3. log | sin x – cos x | + c

4. 

Question 253: