Anti - turunan

Anti - turunan Teorema A Teorema C Teorema B Teorema D

Anti - turunan Teorema A

Anti - turunan Anti - 𝑻𝒖𝒓𝒖𝒏𝒂𝒏 𝑻𝒖𝒓𝒖𝒏𝒂𝒏 𝑦′=𝑓′ 𝑥 𝑦=𝑓 𝑥 𝑦=𝑓 𝑥 𝑦′=𝑓′ 𝑥 4𝑥−5 = 4 2+1 𝑥 1+1 −5𝑥+𝐶 = 2 𝑥 2 −5𝑥+𝐶 = 7 6+1 𝑥 6+1 − 30 4+1 𝑥 4+1 +𝐶 = 𝑥 7 −6 𝑥 5 +𝐶 = 4 3 . 1 1 3 +1 𝑥 4 3 +𝐶 = 𝑥 4 3 +𝐶 𝑻𝒖𝒓𝒖𝒏𝒂𝒏 𝑦′=𝑓′ 𝑥 𝑦=𝑓 𝑥 𝑦=𝑓 𝑥 𝑦′=𝑓′ 𝑥 4𝑥−5 = 4 2+1 𝑥 1+1 −5𝑥+𝐶 = 2 𝑥 2 −5𝑥+𝐶 2 𝑥 2 −5𝑥+3 4𝑥−5 𝑥 7 −6 𝑥 5 −4 7 𝑥 6 −30 𝑥 4 7 𝑥 6 −30 𝑥 4 = 7 6+1 𝑥 6+1 − 30 4+1 𝑥 4+1 +𝐶 = 𝑥 7 −6 𝑥 5 +𝐶 𝑥 4 3 +10 4 3 𝑥 1 3 4 3 𝑥 1 3 = 4 3 . 1 1 3 +1 𝑥 4 3 +𝐶 = 𝑥 4 3 +𝐶

Anti - turunan Teorema B

Teorema C 5 𝑥 2 𝑑𝑥 =5 𝑥 2 𝑑𝑥 Anti - turunan (i) 𝑘𝑓 𝑥 𝑑𝑥=𝑘 𝑓 𝑥 𝑑𝑥 Teorema C memiliki 3 sifat : (i) 𝑘𝑓 𝑥 𝑑𝑥=𝑘 𝑓 𝑥 𝑑𝑥 Contoh : 5 𝑥 2 𝑑𝑥 =5 𝑥 2 𝑑𝑥

Teorema C Anti - turunan (4𝑥+5)𝑑𝑥 = 4𝑥𝑑𝑥 + 5𝑑𝑥 (ii) 𝑓 𝑥 +𝑔 𝑥 𝑑𝑥 = 𝑓 𝑥 𝑑𝑥 + 𝑔 𝑥 𝑑𝑥 Contoh : (4𝑥+5)𝑑𝑥 = 4𝑥𝑑𝑥 + 5𝑑𝑥

Teorema C Anti - turunan (4𝑥−5)𝑑𝑥 = 4𝑥𝑑𝑥 − 5𝑑𝑥 𝑓 𝑥 −𝑔 𝑥 𝑑𝑥 = 𝑓 𝑥 𝑑𝑥 − 𝑔 𝑥 𝑑𝑥 (iii) Contoh : (4𝑥−5)𝑑𝑥 = 4𝑥𝑑𝑥 − 5𝑑𝑥

Anti - turunan Teorema D 𝑔 𝑥 𝑟 .𝑔 𝑥 𝑑𝑥 = 1 𝑟+1 𝑔 𝑥 𝑟+1 +𝐶

Teorema D Anti - turunan Contoh soal : Misal 𝑥 3 +6𝑥 5 . 6 𝑥 2 +12 𝑑𝑥 𝑥 3 +6𝑥 5 . 6 𝑥 2 +12 𝑑𝑥 Misal 𝑈 5 2𝑑𝑢 𝑈= 𝑥 3 +6𝑥 𝑘𝑓 𝑥 𝑑𝑥=𝑘 𝑓 𝑥 𝑑𝑥 =2 𝑈 5 𝑑𝑢 𝑑𝑢 𝑑𝑥 =3 𝑥 2 +6 =2. 1 6 𝑈 6 +𝐶 𝑑𝑢=(3 𝑥 2 +6)𝑑𝑥 3 = 1 3 . 𝑥 3 +6𝑥 6 +𝐶 2𝑑𝑢=2 3 𝑥 2 +6 𝑑𝑥

Latihan Soal dan Soal Bonus

Latihan Soal : Anti - turunan =3. 1 −2+1 𝑥 −2+1 −2. 1 −3+1 𝑥 −3+1 +C 𝑓 𝑥 = 3 𝑥 2 − 2 𝑥 3 =3. 1 −2+1 𝑥 −2+1 −2. 1 −3+1 𝑥 −3+1 +C 3 𝑥 2 − 2 𝑥 3 𝑑𝑥 =−3 𝑥 −1 + 𝑥 −2 +C = (3 𝑥 −2 −2 𝑥 −3 )𝑑𝑥 = −3 𝑥 + 1 𝑥 2 +𝐶

Soal Bonus: Anti - turunan 𝑓 𝑥 = 2𝑥 𝑥 + 3 𝑥 5 𝑓 𝑥 = 2𝑥 𝑥 + 3 𝑥 5 = 2 . 1 −1 2 +1 𝑥 − 1 2 +1 +3 . 1 −5+1 𝑥 −5+1 +𝐶 𝑓 𝑥 = 2 𝑥 1 2 𝑥 + 3 𝑥 5 = 2 . 1 1 2 𝑥 1 2 − 3 4 𝑥 −4 +𝐶 𝑓 𝑥 = 2 𝑥 − 1 2 +3 𝑥 −5 =2 2 𝑥 1 2 − 3 4 𝑥 4 +𝐶 2 𝑥 − 1 2 +3 𝑥 −5 𝑑𝑥

Latihan Soal : Anti - turunan 𝑓 𝑥 = 4 𝑥 6 +3 𝑥 4 𝑥 3 𝑓 𝑥 = 4 𝑥 6 +3 𝑥 4 𝑥 3 =4. 1 3+1 𝑥 3+1 +3. 1 1+1 𝑥 1+1 +𝐶 𝑓 𝑥 = 𝑥 3 4 𝑥 3 +3𝑥 𝑥 3 =4. 1 4 𝑥 4 +3. 1 2 𝑥 2 +𝐶 = 𝑥 4 + 3 2 𝑥 2 +𝐶 𝑓 𝑥 =4 𝑥 3 +3𝑥 = (4 𝑥 3 +3𝑥)𝑑𝑥

Soal Bonus: Anti - turunan 𝑓 𝑥 = 𝑥 6 −𝑥 𝑥 3 𝑓 𝑥 = 𝑥 6 −𝑥 𝑥 3 = 1 3+1 𝑥 3+1 − 1 −2+1 𝑥 −2+1 +𝐶 𝑓 𝑥 = 𝑥 6 𝑥 3 − 𝑥 𝑥 3 = 1 4 𝑥 4 + 𝑥 −1 +𝐶 = 1 4 𝑥 4 + 1 𝑥 +𝐶 𝑓 𝑥 = 𝑥 3 − 𝑥 −2 = ( 𝑥 3 − 𝑥 −2 )𝑑𝑥

Latihan Soal : Anti - turunan sin𝑥 1+cos𝑥 4 𝑑𝑥 sin𝑥 . 𝑈 4 .𝑑𝑥 Misal : = sin𝑥 . 𝑈 4 . 𝑑𝑢 −sin𝑥 𝑈=1+cos𝑥 𝑑𝑢 𝑑𝑥 =−sin𝑥 = − 𝑈 4 𝑑𝑢 𝑑𝑥= 𝑑𝑢 −sin𝑥 =− 1 5 𝑈 5 +𝐶

Soal Bonus: Anti - turunan sin𝑥cos𝑥 . 𝑈 1 2 . 𝑑𝑥 Misal : sin𝑥cos𝑥 1+ sin 2 𝑥 𝑑𝑥 = 1 2 . 1 3 2 𝑈 3 2 +𝐶 sin𝑥cos𝑥 . 𝑈 1 2 . 𝑑𝑥 Misal : = 1 2 . 2 3 𝑈 3 2 +𝐶 sin𝑥cos𝑥 . 𝑈 1 2 . 𝑑𝑢 2sin𝑥cos𝑥 𝑈=1+ sin 2 𝑥 𝑑𝑢 𝑑𝑥 =2sin𝑥cos𝑥 = 1 2 𝑈 1 2 𝑑𝑢 = 1 3 𝑈 3 2 +𝐶 𝑑𝑥= 𝑑𝑢 2sin𝑥cos𝑥 = 1 2 . 1 1 2 +1 . 𝑈 1 2 +1 +𝐶 = 1 3 1+ sin 2 𝑥 3 +𝐶

Latihan Soal : Anti - turunan 𝑓′′ 𝑥 = 𝑥 4 +1 𝑥 3 𝑓′′ 𝑥 = 𝑥 4 +1 𝑥 3 𝑓′ 𝑥 = 1 1+1 𝑥 1+1 + 1 −3+1 𝑥 −3+1 + 𝐶 1 𝑓 𝑥 = 𝑥 3 6 + 1 2𝑥 + 𝐶 1 𝑥+ 𝐶 2 𝑓′′ 𝑥 = 𝑥 4 𝑥 3 + 1 𝑥 3 𝑓′ 𝑥 = 1 2 𝑥 2 − 1 2 𝑥 −2 + 𝐶 1 𝑓 𝑥 = 1 2 𝑥 2 − 1 2 𝑥 −2 + 𝐶 1 𝑑𝑥 𝑓′′ 𝑥 =𝑥+ 𝑥 −3 𝑓′ 𝑥 = (𝑥+ 𝑥 −3 )𝑑𝑥 𝑓 𝑥 = 1 2 . 1 2+1 𝑥 2+1 − 1 2 . 1 −2+1 𝑥 −2+1 + 𝐶 1 𝑥+ 𝐶 2

Soal Bonus: Anti - turunan 𝑓′′ 𝑥 =2 3 𝑥+1 𝑓 𝑥 = 3 2 𝑥+1 4 3 + 𝐶 1 𝑑𝑥 𝑓 𝑥 = 3 2 𝑥+1 4 3 + 𝐶 1 𝑑𝑥 𝑓′′ 𝑥 =2 𝑥+1 1 3 𝑓 𝑥 = 3 2 . 1 4 3 +1 𝑥+1 4 3 +1 + 𝐶 1 𝑥+ 𝐶 2 𝑓′ 𝑥 = 2 𝑥+1 1 3 𝑑𝑥 𝑓 𝑥 = 9 14 𝑥+1 7 3 + 𝐶 1 𝑥+ 𝐶 2 𝑓′ 𝑥 =2. 1 1 3 +1 𝑥+1 1 3 +1 + 𝐶 1 𝑓′ 𝑥 = 3 2 𝑥+1 4 3 + 𝐶 1

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