Integrate[x^2 Cos[x], x] + c c + 2 x Cos @ x D + H -2 + x

(1)

Lösungen

1 Ÿ a

Integrate[x^2 Cos[x], x] + c c + 2 x Cos @ x D + H -2 + x

²

L Sin @ x D

Ÿ b

Integrate[Sin[2x] Cos[x/2], x] + c c - 1

€€€€ 3 Cos A 3 x

€€€€€€€€€

2 E - 1

€€€€ 5 Cos A 5 x

€€€€€€€€€

2 E

Ÿ c

Ln[x_]:=Log[x]

Integrate[Ln[2x], x] + c c - x + x Log @ 2 x D

Integrate[Ln[2x], {x,1,2}]

-1 + Log @ 8 D

N[%]

1.07944

Ÿ d

Integrate[x^3 E^x, x] + c c + ã

^x

H -6 + 6 x - 3 x

²

+ x

³

L

Integrate[x^3 E^x, {x,0,1}]

6 - 2 ã N[%]

0.563436

(2)

Ÿ e

Integrate[Cos[x]^2, x] + c c + €€€€ x

2 + €€€€ 1

4 Sin @ 2 x D

Integrate[Cos[x]^2, {x,0, 2Pi}]

p

N[%]

3.14159

2 Ÿ a

Integrate[x Sqrt[x^2-4], x] + c c + €€€€ 1

3 H -4 + x

²

L

³^•²

Ÿ b

Integrate[Cos[x]^5 Sin[x], x] + c c - 5

€€€€€€€

64 Cos @ 2 x D - 1

€€€€€€€

32 Cos @ 4 x D - 1

€€€€€€€€€€

192 Cos @ 6 x D

Ÿ c

Integrate[x Sqrt[1+x^2], x] + c c + €€€€ 1

3 H 1 + x

²

L

³^•²

Integrate[x Sqrt[1+x^2], {x,0,1}]

€€€€ 1

3 I -1 + 2 •!!!! 2 M N[%]

0.609476

(3)

Ÿ d

Integrate[Sqrt[4-x^2], x] + c c + €€€€ 1

2 x •!!!!!!!!!!!!!! 4 - x

²

+ 2 ArcSin A €€€€ x 2 E

Ÿ e

Integrate[1/Sqrt[4+x^2], x] + c c + ArcSinh A x

€€€€ 2 E

Integrate[1/Sqrt[4+x^2], {x,0, 1}]

ArcSinh A 1

€€€€ 2 E

N[%]

0.481212

3 Integrate[Cos[a x] Sin[a x],{x,0,b}]^2 Sin @ a b D

⁴

€€€€€€€€€€€€€€€€€€€€€€€€€

4 a

²

Integrate[Cos[a x]^2,{x,0,b}]

2 a b + Sin @ 2 a b D

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€

4 a

Integrate[Sin[a x]^2,{x,0,b}]

€€€€ b

2 - €€€€€€€€€€€€€€€€€€€€€€€€€€€ Sin @ 2 a b D 4 a

**Integrate[Cos[a x]^2,{x,0,b}]*Integrate[Sin[a x]^2,{x,0,b}] //Expand** b

²

€€€€€€€

4 - €€€€€€€€€€€€€€€€€€€€€€€€€€€€€ Sin @ 2 a b D

²

16 a

²

4 Ÿ a

Integrate[x^2/(x^3-7),x] + c c + 1

€€€€ 3 Log @ -7 + x

³

D

(4)

Ÿ b

Integrate[E^{2x}/(E^{2x}+7),x] + c 9 c + €€€€ 1

2 Log @ 7 + ã

^{2 x}

D=

Ÿ c

Integrate[(2 Cos[2x] + 4x + E^(-x)) /(Sin[2x] + 2x^2 - E^(-x) + 4),x] + c c - x + Log @-1 + 2 ã

^x

H 2 + x

²

L + ã

^x

Sin @ 2 x DD

Ÿ d

Integrate[Cot[x],x] + c //Expand c + Log @ Sin @ x DD

5 Ÿ a

Apart[1/(x(x-1))]

€€€€€€€€€€€€€€€€ -1 1 + x - 1

€€€€ x

Integrate[1/(x(x-1)),x] + c c + Log @-1 + x D - Log @ x D

Ÿ b

Apart[(x^4-16 x-3)/((x^2+1)(x-1)^2 (x+2))]

- 3

€€€€€€€€€€€€€€€€€€€€€€€€

H -1 + x L

²

+ 2

€€€€€€€€€€€€€€€€ -1 + x + 1

€€€€€€€€€€€€

2 + x + 3 - 2 x

€€€€€€€€€€€€€€€€€

1 + x

²

Integrate[(x^4-16 x-3)/((x^2+1)(x-1)^2 (x+2)),x] + c c + 3

€€€€€€€€€€€€€€€€

-1 + x + 3 ArcTan @ x D + 2 Log @-1 + x D + Log @ 2 + x D - Log @ 1 + x

²

D

(5)

Ÿ c

Apart[(x^4-16 x-3)/((x^2+1)(x-1)^2)]

1 - €€€€€€€€€€€€€€€€€€€€€€€€ 9

H-1 + x L

²

+ €€€€€€€€€€€€€€€€ 3

-1 + x + €€€€€€€€€€€€€€€ 8 - x 1 + x

²

Integrate[(x^4-16 x-3)/((x^2+1)(x-1)^2),x] + c c + 9

€€€€€€€€€€€€€€€€ -1 + x + x + 8 ArcTan @ x D + 3 Log @-1 + x D - 1

€€€€ 2 Log @ 1 + x

²

D

Ÿ d

Apart[(x^2-6 x-6)/(x^2(x^2+3 x+3))]

- 2

€€€€€€€

x

²

+ 3

€€€€€€€€€€€€€€€€€€€€€€€€€€€