1.COS的X的三次方乘以SINX的平方的不定积分是多少? 2.(sint-cost)/(sint+cost)的不定积分

题目

1.COS的X的三次方乘以SINX的平方的不定积分是多少? 2.(sint-cost)/(sint+cost)的不定积分
3.1/(1-cosx)的不定积分
4.根号下（cost+1）除以sinx的平方的不定积分

答案

∫ cos³xsin²x dx
= ∫ cos²xsin²x dsinx
= ∫ (1 - sin²x)sin²x dsinx
= ∫ (sin²x - sin⁴x) dsinx
= (1/3)sin³x - (1/5)sin⁵x + C
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∫ (sint - cost)/(sint + cost) dt
= ∫ d(- cost - sint)/(sint + cost) dt
= - ∫ d(sint + cost)/(sint + cost)
= - ln|sint + cost| + C
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∫ dx/(1 - cosx)
= ∫ 1/(1 - cosx) • (1 + cosx)/(1 + cosx) dx
= ∫ (1 + cosx)/sin²x dx
= ∫ (csc²x + cscxcotx) dx
= - cotx - cscx + C
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∫ √(cosx + 1)/sin²x dx
= 2∫ √[2cos²(x/2) - 1 + 1]/[2sin(x/2)cos(x/2)]² dx/2
= (√2/2)∫ cos(x/2)/[sin²(x/2)cos²(x/2)] dx/2
= (1/√2)∫ csc²(x/2)sec(x/2) dx/2
= (1/√2)∫ [1 + cot²(x/2)]sec(x/2) dx/2
= (1/√2)∫ sec(x/2) dx/2 + (1/√2)∫ csc(x/2)cot(x/2) dx/2
= (1/√2)ln| sec(x/2) + tan(x/2) | - (1/√2)csc(x/2) + C