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exercise for seriess, Exams of Fourier Transform and Series

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2022/2023

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ANAL˙
IZ II
UYGULAMALAR
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ANAL˙IZ II

UYGULAMALAR

Ornek 1¨

I =

∫ (^) π

0

x sin xdx

1 + cos 2x

integralini hesaplayınız.

C. ¨oz¨um¨un Devamı : O halde u = cos t d¨on¨u¸s¨um¨u yapılarak

I =

π

2

− 1

du

1 + u^2

π

2

arctan u

1

− 1

π^2

4

elde edilir.

Ornek 2∫¨

π

−π

x − sin x

(x^2 + 2 cos x)^2

dx integralini hesaplayınız.

Ornek 2∫¨

π

−π

x − sin x

(x^2 + 2 cos x)^2

dx integralini hesaplayınız.

C. ¨oz¨um : ˙Integrasyon aralı˘gı simetrik olup integrand

f (−x) =

−x − sin(−x)

((−x)^2 + 2 cos(−x))^2

x − sin x

(x^2 + 2 cos x)^2

= −f (x)

oldu˘gundan tek fonksiyondur. O halde

∫ (^) π

−π

x − sin x

(x^2 + 2 cos x)^2

dx = 0

olur.

Ornek 3¨

E˘ger x > 2 ise,

∫ (^) x 2

2

(t − 2)dt

t^2

t − 1

denklemini c.¨oz¨un¨uz.

Ornek 4¨

0

x arcsin(x^2 )dx integralini hesaplayınız.

Ornek 4¨

0

x arcsin(x^2 )dx integralini hesaplayınız.

C. ¨oz¨um : u = arcsin(x^2 ) ve dv = xdx alınıp kısmi integrasyon

kuralı uygulandı˘gında verilen integral

∫ (^1)

0

x arcsin(x

2 )dx =

x

2 arcsin(x

2 )

1

0

0

x^3 dx √ 1 − x^4

halini alır.

Ornek 5¨

1 − x^4

x^3

dx integralini hesaplayınız.

Ornek 5¨

1 − x^4

x^3

dx integralini hesaplayınız.

C. ¨oz¨um : x^2 = sin t d¨on¨u¸s¨um¨u uygulandı˘gında verilen integralden

I =

1 − sin

2 t

(sin t)

3 2

cos t √ sin t

dt =

cos^2 t

sin^2 t

dt =

cot

2 tdt

(cot t + t) + C = −

1 − x^4

2 x^2

arcsin(x^2 ) + C

elde edilir.

Ornek 6¨

I =

dx √ 5 − 12 x − 9 x^2

ibtegralini hesaplayınız.

C. ¨oz¨um : 5 − 12 x − 9 x^2 = 9 − (3x + 2)^2 ¸seklinde

yazılabilece˘ginden

Ornek 6¨

I =

dx √ 5 − 12 x − 9 x^2

ibtegralini hesaplayınız.

C. ¨oz¨um : 5 − 12 x − 9 x^2 = 9 − (3x + 2)^2 ¸seklinde

yazılabilece˘ginden 3 sin t = 3x + 2 d¨on¨u¸s¨um¨u uygulandı˘gında

I =

cos tdt √ 9 − 9 sin^2 t

dt =

arcsin t + C

arcsin

3 x + 2

3

+ C

bulunur.

Ornek 7∫¨

1 − exdx integralini hesaplayınız.

C. ¨oz¨um : t^2 = 1 − ex^ d¨on¨u¸s¨um¨u yapıldı˘gında

2 tdt = −e

x dx ⇒

2 tdt

t^2 − 1

= dx olup verilen integral

I =

2 t^2 dt

t^2 − 1

t^2 − 1

dt =

t − 1

t + 1

dt

= 2 t + ln |t − 1 | + ln |t + 1| + C

= 2

1 − ex^ + ln |

1 − ex^ − 1 | − ln |

1 − ex^ + 1| + C

oldu˘gu g¨or¨ul¨ur.

Ornek 8¨

y = 2 − x do˘grusu, y = 4 − x^2 parabolu ve x-ekseni ile

sınırlandırılmı¸s b¨olgenin alanını bulunuz.