3. 2. 6. x log − 5. 1 e ( sen x + cos x )2 1.

Calcolo integrale – soluzioni degli esercizi proposti N. 1

N.B.: Tutte le soluzioni sono date a meno di una costante additiva arbitraria

1.

1. ^{3 x} 1 ² 2 2

e x - x +

3 9 27

⎛ ⎞

⎜ ⎟

⎝ ⎠ 2. x tg x + log cosx

3.

(x + 1 ) arctg x - x 2

2 4. 1 ^{- x}

e ( sen x + cos x )

− 2

5. x log ( x + 1 + x ) - 1 + x ^{2 2} 6. x log ² x – 2x ( log x – 1 )

7.

4 4

x logx x

4 - 16

8.

3 2

x 3 x 3x 3

- sen 2x + - cos 2x

2 4 4 8

⎛ ⎞ ⎛ ⎞

⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

2.

1. sen x – ( sen ³ x ) / 3 2. arctg ( x ² ) / 2

3. log log x 4. log ( 1 + e ^x )

5. arctg ( log x ) - ½ log ( 1 + log ² x ) 6. 1 ^{2 3/2} - ( 2 - x )

3 7. – arctg cos x

3.

1.

x x

e + 1 - 1 log

e + 1 + 1

2.

x + 1 2

− x

3. 2 1 + sen x 4. x + 4 1 + x + 4 log 1 + x - 1

5. ^{1 1} ₂

4 2

1 - cos x cos x

log -

1 + cos x sen x 6. x

tg 2

7. tg x - x 8.

x x 4 - x 2

2 arcsen +

2 2

9. 2 x + 1

arcsen

3 10.

1 - x 1 -

1 - x 1 + x

log + 2 arctg

1 + x 1 - x

1 + 1 + x

4.

1. ^{2 2 1} ₂ x

x - 4 log ( x + 4 ) + arctg

2 2. 3 log x - 3 - 2 log x - 2

3. ^{1 2} 3

2 3

x - 1

log ( x - 2 x + 4 ) + arctg 4.

2

x - 2 2

log - arctg x - x - 2 x + 1

5. ^{2 10}

31 31

4 x - 5 2 log ( 2 x - 5 x + 7 ) + arctg

6. ^{1 1}

2 2

x - 1

log - arctg x

x + 1 5.

1. 2

1 - tg ( x / 2 ) 2. log tg x

3.

e tg x

2 4. - log x - 4 x + 5 - x + 2 ²

5.

2 2

x - 2 x - x log

x + 2 x - x 6. tg x - ^{1 2}

2

log ( tg x + 2 ) + 2 arctg tg x 2

7.

x 1

1 2 2

5 x

2 + 2

tg - log

tg 8. ¹

2

tg x - 1 log

tg x + 1 9.

2 1 1 2

4 4

2

x arcsen x - arcsen x + x 1 - x