第12类RD Sharma解–第19章不定积分–练习19.23 |套装2

问题9.评估∫1 / cosx + sinx dx

解决方案：

Let us assume I = ∫ 1/ cosx+sinx dx

Put sinx = 2tan(x/2)/ 1+tan²x/2

cosx = 1-tan²(x/2)/1+tan²(x/2)

= ∫ 1/ {1-tan²(x/2)/1+tan²x/2} + {2tan(x/2)/1+tan²x/2} dx

= ∫ 1+tan²(x/2)/ 2tan(x/2)+1-tan²(x/2) dx

= ∫ sec²(x/2)/ 2tan(x/2)+1-tan²(x/2) dx (i)

Let tanx/2 = t

1/2 sec²x/2 dx = dt

Put the above value in eq. (i)

= ∫ 2dt/ 2t+1-t²

= -∫ 2dt/ t²-2t-1

= -∫ 2dt/ t²-2t+1-1-1

= -∫ 2dt/ (t-1)²-(√2)²

= ∫ 2dt/ (√2)²-(t-1)²

Integrate the above eq. then, we get

= 2/(2√2) log|√2+t-1/√2-t+1| +c

Hence, I = 1/√2 log|√2+tanx/2-1/ √2-tanx/2+1| +c

问题10.评估∫1 / 5-4cosx dx

解决方案：

Let us assume I = ∫ 1/ 5-4cosx dx

Put cosx = 1-tan²(x/2)/ 1+tan²(x/2)

= ∫1/ 5-4{1-tan²(x/2)/ 1+tan²(x/2)} dx

= ∫ 1+tan²(x/2)/ 5(1+tan²x/2)-4(1-tan²x/2) dx

= ∫ sec²(x/2)/ 5+5tan²x/2-4+4tan²x/2 dx

= ∫ sec²(x/2)/ 9tan²x/2+1 dx (i)

Let 3tanx/2 = t

3/2 sec²x/2 dx = dt

Put the above value in eq. (i)

= 2/3 ∫ dt/ t²+1

On integrate the above eq. then, we get

= 2 × 1/3tan^-1t +c

= 2/3 tan^-1{3tan(x/2)} +c

Hence, I = 2/3 tan^-1{3tan(x/2)} +c

问题11.评估∫1/2 + sinx + cosx dx

解决方案：

Let us assume I = ∫ 1/ 2+sinx+cosx dx

Put sinx = 2tan(x/2)/ 1+tan²x/2

cosx = 1-tan²(x/2)/1+tan²(x/2)

= ∫ 1/ 2+{2tan(x/2)/1+tan²x/2} + {1-tan²(x/2)/1+tan²x/2} dx

= ∫ 1+tan²(x/2)/ 2+2tan²(x/2)+2tan(x/2)+1-tan²(x/2) dx

= ∫ sec²(x/2)/ tan²x/2+2tanx/2+3 dx (i)

Let tanx/2 = t

1/2 sec²x/2 dx = dt

Put the above value in eq. (i)

= ∫ 2dt/ t²+2t+3

= ∫ 2dt/ t²+2t+1-1+3

= 2∫ dt/ (t+1)²+(√2)²

Integrate the above eq. then, we get

= 2/√2 tan^-1(t+1/ √2) +c

Hence, I = √2tan^-1(tanx/2+1/ √2) +c

问题12.评估∫1 / sinx +√3cosxdx

解决方案：

Let us assume I = ∫ 1/ sinx+√3cosx dx

Put sinx = 2tan(x/2)/ 1+tan²x/2

cosx = 1-tan²(x/2)/1+tan²(x/2)

= ∫ 1/ {2tan(x/2)/1+tan²x/2}+√3{1-tan²(x/2)/1+tan²x/2} dx

= ∫ 1+tan²(x/2)/ 2tan(x/2)+√3-√3tan²(x/2) dx

= ∫ sec²(x/2)/ 2tan(x/2)+√3-√3tan²(x/2) dx (i)

Let tanx/2 = t

1/2 sec²x/2 dx = dt

Put the above value in eq. (i)

= ∫ 2dt/ 2t+√3-√3t²

= -2/√3 ∫ dt/ t²-2/√3t+(1/√3)²-(1/√3)²-1

= -2/√3 ∫ dt/ (t-1/√3)²-(2/√3)²

= 2/√3 ∫ dt/ (2/√3)²-(t-1/√3)²

Integrate the above eq. then, we get

= 2/√3 x 1/2(2/√3) log|2/√3+t+1/√3/ 2/√3-t+1/√3| +c

= 1/2 log|√3t+1/ 3-√3t|+c

Hence, I = 1/2 log|1+√3+tanx/2/ 3-√3tanx/2| +c

问题13.评估∫1 /√3sinx+ cosx dx

解决方案：

Let us assume I = ∫ 1/ √3sinx+cosx dx

√3 = rcosθ, and 1=rsinθ

tanθ=1/√3

θ = π/6

r = √3+1=2

I = ∫ 1/ rcosθsinx+rsinθcosx dx

= 1/r ∫ 1/ sin(x+θ) dx

= 1/2 ∫ cosec(x+θ) dx

Integrate the above eq. then, we get

= 1/2 log|tan(x/2+θ/2)| +c

Hence, I = 1/2 log|tan(x/2+π/12)| +c

问题14.评估∫1 /sinx-√3cosxdx

解决方案：

Let us assume I = ∫ 1/ sinx-√3cosx dx

1 = rcosθ, and √3=rsinθ

tanθ=√3

θ = π/3

r = √3+1=2

I = ∫ 1/ rcosθsinx-rsinθcosx dx

= 1/r ∫ 1/ sin(x-θ) dx

= 1/2 ∫ cosec(x-θ) dx

Integrate the above eq. then, we get

= 1/2 log|tan(x/2-θ/2)| +c

Hence, I = 1/2 log|tan(x/2-π/6)| +c

问题15.评估∫1/5 + 7cosx + sinx dx

解决方案：

Let us assume I = ∫ 1/ 5+7cosx+sinx dx

Put sinx = 2tan(x/2)/ 1+tan²x/2

cosx = 1-tan²(x/2)/1+tan²(x/2)

= ∫ 1/ 5+7{1-tan²(x/2)/1+tan²(x/2)} + {2tan(x/2)/ 1+tan²x/2} dx

= ∫ 1+tan²(x/2)/ 5(1+tan²x/2)+7-7tan²(x/2)+2tan(x/2) dx

= ∫ sec²(x/2)/ -2tan²x/2+12+2tanx/2 dx (i)

Let tanx/2 = t

1/2 sec²x/2 dx = dt

Put the above value in eq. (i)

= ∫ 2dt/ -2t²+12+2t

= -∫ dt/ t²-t-6

= – ∫ dt/ t²-2t(1/2)+(1/2)²-(1/2)²-6

= – ∫ dt/ (t-1/2)²-(5/2)²

Integrate the above eq. then, we get

= -1/2(5/2) log|t-1/2-5/2/ t-1/2+5/2| +c

= -1/5 log|t-3/ t+2| +c

Hence, I = 1/5 log|tanx/2+2/ tanx/2-3| +c