Integration

A2 Integration II Starter: KUS objectives BAT use integration by parts with trig functions including ‘cyclic’ problems Starter:

WB 7a find the following integrals a) 𝑥 cos 𝑥 𝑑𝑥 b) 2𝑥 sin 3𝑥 𝑑𝑥 𝑢=𝑥 𝑑𝑣 𝑑𝑥 = cos 𝑥 𝑢 𝑑𝑣 𝑑𝑥 = 𝑢𝑣 − 𝑣 𝑑𝑢 𝑑𝑥 𝑣= sin 𝑥 𝑥 cos 𝑥 𝑑𝑥 𝑑𝑢 𝑑𝑥 =1 =𝑥 sin 𝑥 − sin 𝑥 1 𝑑𝑥 =𝑥 sin 𝑥 + cos 𝑥 +𝐶

WB 7b find the following integrals a) 𝒙 𝐜𝐨𝐬 𝒙 𝒅𝒙 b) 𝟐𝒙 𝐬𝐢𝐧 𝟑𝒙 𝒅𝒙 𝑢=2𝑥 𝑑𝑣 𝑑𝑥 = 𝑠𝑖𝑛 3𝑥 𝑢 𝑑𝑣 𝑑𝑥 = 𝑢𝑣 − 𝑣 𝑑𝑢 𝑑𝑥 𝑣=− 1 3 cos 3𝑥 = 2 3 𝑥 cos 𝑥 − − 2 3 cos 3𝑥 𝑑𝑥 𝑑𝑢 𝑑𝑥 =2 = 2 3 𝑥 cos 𝑥 + 2 9 sin 3𝑥 +𝐶

WB 8 Show that 0 𝜋/2 4𝑥 sin 4𝑥 𝑑𝑥 =− 1 8 𝜋 𝑢=4𝑥 𝑣=− 1 4 cos 4𝑥 𝑑𝑢 𝑑𝑥 =4 𝑑𝑣 𝑑𝑥 = sin 4𝑥 𝑢 𝑑𝑣 𝑑𝑥 = 𝑢𝑣 − 𝑣 𝑑𝑢 𝑑𝑥 =− 𝑥 cos 4𝑥 − − cos 4𝑥 𝑑𝑥 = − 1 4 𝑥 cos 4𝑥 + 1 16 sin 4𝑥 𝜋/2 0 = − 1 4 × 𝜋 2 1 + − 0+0 =− 1 8 𝜋 QED

WB 9 Show that 𝜋/6 𝜋/3 3𝑥 cos 3𝑥 𝑑𝑥 =− 1 6 (2+𝜋) 𝑢=3𝑥 𝑣= 1 3 sin 3𝑥 𝑑𝑢 𝑑𝑥 =3 𝑑𝑣 𝑑𝑥 = cos 3𝑥 𝑢 𝑑𝑣 𝑑𝑥 = 𝑢𝑣 − 𝑣 𝑑𝑢 𝑑𝑥 = 𝑥 sin 3𝑥 − sin 3𝑥 𝑑𝑥 = 𝑥 sin 3𝑥 + 1 3 𝑐𝑜𝑠 3𝑥 𝜋/3 𝜋/6 = 𝜋 3 0 + 1 3 −1 − 𝜋 6 1 + 1 3 0 =− 1 3 − 𝜋 6 =− 1 6 (2+𝜋) QED

WB 10 Circular questions Find 𝑒 𝑥 cos 𝑥 𝑑𝑥 𝑢 𝑑𝑣 𝑑𝑥 = 𝑢𝑣 − 𝑣 𝑑𝑢 𝑑𝑥 WB 10 Circular questions Find 𝑒 𝑥 cos 𝑥 𝑑𝑥 𝑢= 𝑒 𝑥 𝑣= sin 𝑥 𝑑𝑢 𝑑𝑥 = 𝑒 𝑥 𝑑𝑣 𝑑𝑥 = cos 𝑥 Round 1 = 𝑒 𝑥 sin 𝑥 − 𝑒 𝑥 sin 𝑥 𝑑𝑥 = 𝑒 𝑥 sin 𝑥 − − 𝑒 𝑥 cos 𝑥 − − 𝑒 𝑥 cos 𝑥 𝑑𝑥 = 𝑒 𝑥 sin 𝑥 + 𝑒 𝑥 cos 𝑥 − 𝑒 𝑥 cos 𝑥 𝑑𝑥 𝑢= 𝑒 𝑥 𝑣=− cos 𝑥 𝑑𝑢 𝑑𝑥 = 𝑒 𝑥 𝑑𝑣 𝑑𝑥 = sin 𝑥 Round 2 How is this circular? 2 𝑒 𝑥 cos 𝑥 𝑑𝑥 = 𝑒 𝑥 sin 𝑥 + 𝑒 𝑥 cos 𝑥 𝑒 𝑥 cos 𝑥 𝑑𝑥 = 1 2 𝑒 𝑥 sin 𝑥 + cos 𝑥 +C

Skills 219 homework 219

One thing to improve is – KUS objectives BAT use integration by parts with trig functions including ‘cyclic’ problems self-assess One thing learned is – One thing to improve is –

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