integration

EXERCISES : INTEGRATION

1. Integrate the following wrt x: 2

5

1

a) âˆŤ (đ?‘Ľ 4 − đ?‘Ľ 3 + 10) đ?‘‘đ?‘Ľ

e) âˆŤ (4đ?‘’ −5đ?‘Ľ − 3 đ?‘’ 2đ?‘Ľ ) đ?‘‘đ?‘Ľ

b) âˆŤ(3đ?‘Ľ 4 + 7đ?‘Ľ 2 − 6) đ?‘‘đ?‘Ľ

f) âˆŤ (đ?‘Ľ − 1−4đ?‘Ľ) đ?‘‘đ?‘Ľ

2

1

3

c) âˆŤ ( 3−4đ?‘Ľ) đ?‘‘đ?‘Ľ √

g) âˆŤ (1+4đ?‘Ľ 2 ) đ?‘‘đ?‘Ľ đ?œ‹

3

d) âˆŤ 3đ?‘?đ?‘œđ?‘ (3đ?‘Ľ − ) đ?‘‘đ?‘Ľ 4

ďƒ˛  2x  1

2. Find 3. Find

ďƒ˛  2  3x 

2

3



2

h) âˆŤ ( ) đ?‘‘đ?‘Ľ √4−đ?‘Ľ 2

1  sec 2 3x dx 2x  1 .



1  cos  2x  1 dx .

3x  2

4. Find âˆŤ(√đ?‘Ľ + 1)(đ?‘Ľ − 2) đ?‘‘đ?‘Ľ 5. Find

ďƒ˛e

2

6. Evaluate

 x

ďƒ˛x

lnx

Evaluate

ďƒ˛

8. Evaluate

ďƒ˛

7.

3 2

1  sin  2x  1 dx .

2x  1

dx using integration by parts.

x ln x dx using integration by parts. e2x ex  1

dx by using appropriate substitution.

ANSWERS: 2

5

1 . a)- 3đ?‘Ľ 3 + 2đ?‘Ľ2 + 10đ?‘Ľ + đ?‘? b) c)

3đ?‘Ľ 5 5

+

7đ?‘Ľ 23 3

√3−4đ?‘Ľ −2

− 6đ?‘Ľ + đ?‘?

+đ?‘?

e) −

đ?œ‹

h)

1

− 6 đ?‘’ 2đ?‘Ľ + đ?‘?

f) 2 �� � + g)

d) sin (3đ?‘Ľ − 4 ) + đ?‘?

4đ?‘’ −5đ?‘Ľ 5

3 đ?‘™đ?‘›(1−4đ?‘Ľ) 4

3 tan−1 (2đ?‘Ľ) + 2

đ?‘?

3 1 sin−1 (2 đ?‘Ľ) + đ?‘?

4

+đ?‘?

EXERCISES : INTEGRATION 2.

3.

4.

5.

6.

(2𝑥+1)3 6

+

(2−3𝑥)−1

−

3 2 5 𝑥2 5

𝑙𝑛(2𝑥+1) tan(3𝑥) − + 2 3

𝑙𝑛(3𝑥+2) sin(2𝑥+1) − + 3 2

1

4

𝑐

3

+ 2 𝑥 2 − 3 𝑥 2 − 2𝑥 + 𝑐 2

3

4 √𝑥

+𝑐

𝑒2𝑥 + 3 𝑥2 + −2𝑙𝑛𝑥 √𝑥

𝑐

−

7.

2 3 𝑥 2 𝑙𝑛𝑥 3

8.

2 (𝑒 𝑥 3

4

𝑙𝑛(2𝑥+1) cos(2𝑥+1) − 2 2

3

− 9 𝑥2 + 𝑐 3

1

+ 1)2 − 2(𝑒 𝑥 + 1)2 + 𝑐

+𝑐