高数不定积分帮求下不定积分，要详
1。∫cos^3 θ*sinθdθ
=-∫cos^3 θd(cosθ)
=(-1/4)cos^4 θ+C
2。∫[e^x/(e^2x+4)]dx
=∫[1/(e^x)^2+2^2]d(e^x)
=(1/2)actan(e^x/2)+C【这个直接有公式】
3。 ∫ln(1+x^2)dx
=x*ln(1+x^2)-∫x*d[ln(1+x^2)]【分部积分法】
=x*ln(1+x^2)-∫x*[2x/(1+x^2)]dx
=x*ln(1+x^2)-2∫[x^2/(1+x^2)]dx
=x*ln(1+x^2)-2∫[(x^2+1-1)/(1+x^2)]dx
=x*ln(1+x^2)-2∫dx+2...全部
1。∫cos^3 θ*sinθdθ
=-∫cos^3 θd(cosθ)
=(-1/4)cos^4 θ+C
2。∫[e^x/(e^2x+4)]dx
=∫[1/(e^x)^2+2^2]d(e^x)
=(1/2)actan(e^x/2)+C【这个直接有公式】
3。
∫ln(1+x^2)dx
=x*ln(1+x^2)-∫x*d[ln(1+x^2)]【分部积分法】
=x*ln(1+x^2)-∫x*[2x/(1+x^2)]dx
=x*ln(1+x^2)-2∫[x^2/(1+x^2)]dx
=x*ln(1+x^2)-2∫[(x^2+1-1)/(1+x^2)]dx
=x*ln(1+x^2)-2∫dx+2∫[1/(x^2+1)]dx
=x*ln(1+x^2)-2x+2arctanx+C
4。
∫x^5*sin(x^2)dx
=∫(1/2)*x^4*sin(x^2)d(x^2)
=-(1/2)∫x^4d[cos(x^2)]
=(-1/2)[x^4*cos(x^2)-∫cos(x^2)d(x^4)]
=(-1/2)[x^4*cos(x^2)-∫4x^3*cos(x^2)dx]
=(-1/2)x^4*cos(x^2)+∫2x^3*cos(x^2)dx
=(-1/2)x^4*cos(x^2)+∫x^2*cos(x^2)d(x^2)
=(-1/2)x^4*cos(x^2)+∫x^2d[sin(x^2)]
=(-1/2)x^4*cos(x^2)+[x^2*sin(x^2)-∫sin(x^2)d(x^2)]
=(-1/2)x^4*cos(x^2)+x^2*sin(x^2)+cos(x^2)+C
5。
∫(x^2+7x-5)cos2xdx
=(1/2)∫(x^2+7x-5)d(sin2x)
=(1/2)[(x^2+7x-5)*sin2x-∫sin2xd(x^2+7x-5)]
=(1/2)(x^2+7x-5)*sin2x-(1/2)∫sin2x*(2x+7)dx
=(1/2)(x^2+7x-5)*sin2x+(1/2)∫(1/2)(2x+7)d(cos2x)
=(1/2)(x^2+7x-5)*sin2x+(1/4)[(2x+7)cos2x-∫cos2xd(2x+7)]
=(1/2)(x^2+7x-5)*sin2x+(1/4)(2x+7)cos2x-(1/4)∫cos2xd(2x)
=(1/2)(x^2+7x-5)*sin2x+(1/4)(2x+7)cos2x+(1/4)sin2x+C。
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