Integration

Integration I Starter: KUS objectives BAT Integrate using a substitution BAT find a definite integral using a substitution, including converting the bounds Starter:

WB 9a Find tan 𝑥 𝑑𝑥 using the substitution 𝑢= cos 𝑥 To integrate this, you need to replace the x terms with equivalent u terms, and replace the dx with an equivalent du = sin 𝑥 cos 𝑥 𝑑𝑥 𝒖=𝒄𝒐𝒔 𝒙 𝒙=𝐮 −𝟏 =− 1 𝑢 𝑑𝑢 𝒅𝒖 𝒅𝒙 =−𝒔𝒊𝒏 𝒙 𝒅𝒖=−𝒔𝒊𝒏 𝒙 𝒅𝒙 =− ln cos 𝑥 +𝐶 = ln ( cos 𝑥) −1 +𝐶 = ln sec 𝑥 +𝐶

WB 9b Find cot 𝑥 𝑑𝑥 = cos 𝑥 sin 𝑥 𝑑𝑥 𝒖= 𝒔𝒊𝒏 𝒙 = 1 𝑢 𝑑𝑢 𝒅𝒖 𝒅𝒙 =𝒄𝒐𝒔 𝒙 = 1 𝑢 𝑑𝑢 𝒅𝒖 𝒅𝒙 =𝒄𝒐𝒔 𝒙 = ln 𝑢 +𝐶 𝒅𝒖=𝒄𝒐𝒔 𝒙 𝒅𝒙 = ln sin 𝑥 +𝐶

WB 10 Find cos 𝑥 sin 𝑥 1+ sin 𝑥 3 𝑑𝑥 using the substitution 𝑢= sin 𝑥 +1 sin 𝑥 1+ sin 𝑥 3 cos 𝑥 𝑑𝑥 𝒖= 𝒔𝒊𝒏 𝒙+𝟏 = 𝑢−1 𝑢 3 𝑑𝑢 𝒖−𝟏=𝒔𝒊𝒏 𝒙 = 𝑢 4 − 𝑢 3 𝑑𝑢 𝒅𝒖 𝒅𝒙 =𝒄𝒐𝒔 𝒙 = 1 5 𝑢 5 − 1 4 𝑢 4 +𝐶 𝒅𝒖=𝒄𝒐𝒔 𝒙 𝒅𝒙 = 1 5 𝒔𝒊𝒏 𝒙+𝟏 5 − 1 4 𝒔𝒊𝒏 𝒙+𝟏 4 +𝐶

WB 11 Show that 0 𝜋/2 cos 𝑥 3+2 sin 𝑥 𝑑𝑥 = 1 2 ln 5 3 = 1 3+2 sin 𝑥 2 cos 𝑥 𝑑𝑥 𝒖= 𝟑+𝟐 𝒔𝒊𝒏 𝒙 = 1 2 1 𝑢 𝑑𝑢 𝒅𝒖 𝒅𝒙 =𝟐 𝒄𝒐𝒔 𝒙 = 1 2 ln 𝑢 +𝐶 𝒅𝒖=𝟐 𝒄𝒐𝒔 𝒙 𝒅𝒙 = 1 2 ln 3+2 sin 𝑥 = 1 2 ln (3+2) − 1 2 ln (3+0 = 1 2 ln 5 3

WB 12 Find cos 𝑥 𝑠𝑖𝑛 5 𝑥 𝑑𝑥 = 𝑠𝑖𝑛 5 𝑥 cos 𝑥 𝑑𝑥 𝒖= 𝒔𝒊𝒏 𝒙 = 𝑢 5 𝑑𝑢 𝒖= 𝒔𝒊𝒏 𝒙 = 𝑢 5 𝑑𝑢 𝒅𝒖 𝒅𝒙 =𝒄𝒐𝒔 𝒙 = 1 6 𝑢 6 +𝐶 𝒅𝒖= 𝒄𝒐𝒔 𝒙 𝒅𝒙 = 1 6 𝑠𝑖𝑛 6 𝑥 +𝐶

WB 13 Find sin 𝑥 𝑐𝑜𝑠 𝑛 𝑥 𝑑𝑥 = 𝑐𝑜𝑠 𝑛 𝑥 sin 𝑥 𝑑𝑥 𝒖= 𝒄𝒐𝒔 𝒙 =− 𝑢 𝑛 𝑑𝑢 𝒖= 𝒄𝒐𝒔 𝒙 =− 𝑢 𝑛 𝑑𝑢 𝒅𝒖 𝒅𝒙 =− 𝒔𝒊𝒏 𝒙 =− 1 𝑛+1 𝑢 6 +𝐶 𝒅𝒖=− 𝒔𝒊𝒏 𝒙 𝒅𝒙 = 1 𝑛+1 𝑐𝑜𝑠 𝑛+1 𝑥 +𝐶

WB 14 Find 𝑐𝑜𝑠𝑒𝑐 𝑥 𝑑𝑥 = 𝑐𝑜𝑠𝑒𝑐 2 𝑥+𝑐𝑜𝑠𝑒𝑐 𝑥 cot 𝑥 cosec 𝑥 + cot 𝑥 𝑑𝑥 Multiply through by 𝑐𝑜𝑠𝑒𝑐 𝑥+ cot 𝑥 cosec 𝑥 + cot 𝑥 =− 1 𝑢 𝑑𝑢 𝒖= 𝒄𝒐𝒔𝒆𝒄 𝒙+𝒄𝒐𝒕 𝒙 =− ln 𝑢 +𝐶 𝒅𝒖 𝒅𝒙 =−𝒄𝒐𝒔𝒆𝒄 𝒙 𝒄𝒐𝒕 𝒙 − 𝒄𝒐𝒔𝒆𝒄 𝟐 𝒙 =− ln 𝒄𝒐𝒔𝒆𝒄 𝒙+𝒄𝒐𝒕 𝒙 +𝐶 𝒅𝒖=−𝒄𝒐𝒔𝒆𝒄 𝒙 𝒄𝒐𝒕 𝒙 − 𝒄𝒐𝒔𝒆𝒄 𝟐 𝒙 dx

WB 15 Find 𝑠𝑒𝑐 𝑥 𝑑𝑥 = 𝑠𝑒𝑐 2 𝑥+𝑠𝑒𝑐 𝑥 tan 𝑥 sec 𝑥 + tan 𝑥 𝑑𝑥 Multiply through by 𝑠𝑒𝑐 𝑥+ tan 𝑥 sec 𝑥 + tan 𝑥 = 1 𝑢 𝑑𝑢 𝒖= 𝒔𝒆𝒄 𝒙+𝒕𝒂𝒏 𝒙 = ln 𝑢 +𝐶 𝒅𝒖 𝒅𝒙 =𝒔𝒆𝒄 𝒙 𝒕𝒂𝒏 𝒙+ 𝒔𝒆𝒄 𝟐 𝒙 = ln 𝒔𝒆𝒄 𝒙+𝒕𝒂𝒏 𝒙 +𝐶 𝒅𝒖=𝒔𝒆𝒄 𝒙 𝒕𝒂𝒏 𝒙+ 𝒔𝒆𝒄 𝟐 𝒙 dx

The formula booklet gives you some help We need more techniques before meeting all of these

KUS objectives Bat Integrate using a substitution BAT find a definite integral using a substitution, including converting the bounds self-assess One thing learned is – One thing to improve is –

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