问题11.评估∫sin2x /√cos4×-SIN 2×+ 2 DX
解决方案:
Let us assume I =∫ sin2x/ √cos4x-sin2x+2 dx
=∫ sin2x/ √cos4x-(1-cos2x)+2 dx (i)
Put cos2x = t
-2sinxcosx dx = dt
sin2x dx = -dt
Put the above value in eq. (i)
= -∫ dt/ √t2-(1-t)+2
= -∫ dt/ √t2-1+t+2
= -∫ dt/ √t2+t+1
= -∫ dt/ √t2+t+(1/4)+(3/4)
= -∫ dt/ √(t+1/2)2+ 3/4 (ii)
Put t+1/2 =u
dt = du
Put the above value in eq. (ii)
= -∫ du/ √u2+ 3/4
= -∫ du/ √u2+3/4
Integrate the above eq. then, we get
= -log|u +√u2+3/4| + c [since ∫ 1/√x2+a2 dx =log|x +√x2+a2| + c]
= -log|t+1/2 +√(t+1/2)2+3/4| + c
= -log|t+1/2 +√(t2+t+1)| + c
= -log|(cos2x+1/2) +√(cos4x+cos2x+1| + c
Hence, I = -log|(cos2x+1/2) +√(cos4x+cos2x+1| + c
问题12评估∫cosx /√4-SIN 2×DX
解决方案:
Let us assume I =∫ cosx/ √4-sin2x dx (i)
Put sinx = t
cosx dx = dt
Put the above value in eq. (i)
= ∫ dt/ √(2)2-(t)2
Integrate the above eq. then, we get
= sin-1(t/2) + c [ since ∫1/ √a2 – x2 dx = sin-1(x/a) + c]
= sin-1(sinx/2) + c
Hence, I = sin-1(sinx/2) + c
问题13.评估∫1 / x 2/ 3√x2/3 -4 dx
解决方案:
Let us assume I =∫ 1/ x2/3√x2/3-4 dx (i)
Put x1/3 = t
(1/3) x1/3-1 dx = dt
(1/3) x-2/3 dx = dt
dx/ x2/3 = 3dt
Put the above value in eq. (i)
= 3 ∫ dt/ √t2-(2)2
Integrate the above eq. then, we get
= 3 log|t +√t2-(2)2| + c [since ∫ 1/√x2-a2 dx =log|x +√x2-a2| + c]
= 3 log|x1/3 +√(x1/3)2-(2)2| + c
Hence, I = 3 log|x1/3 +√x2/3-4| + c
问题14.评估∫1 /√(1-x 2 )[9+(sin -1 x) 2 dx
解决方案:
Let us assume I =∫ 1/ √(1-x2)[9+(sin-1x)2 dx (i)
Put sin-1x = t
dx/√1-x2 = dt
Put the above value in eq. (i)
= ∫ dt/ √(3)2+t
Integrate the above eq. then, we get
= log|t +√(3)2+t2| + c [since ∫ 1/√a2+x2 dx =log|x +√a2+x2| + c]
= log|sin-1x +√(3)2+(sin-1x)2| + c
Hence, I = log|sin-1x +√9+(sin-1x)2| + c
问题15.评估∫cosx /√sin2×-2sinx-3 DX
解决方案:
Let us assume I =∫ cosx/ √sin2x-2sinx-3 dx (i)
Put sinx = t
cosx dx = dt
Put the above value in eq. (i)
= ∫ dt/ √t2-2t-3
= ∫ dt/ √t2-2t+(1)2-(1)2-3
= ∫ dt/ √(t-1)2-(2)2 (ii)
Put t-1 =u
dt = du
Put the above value in eq. (ii)
= ∫ du/ √u2-(2)2
Integrate the above eq. then, we get
= log|u +√u2-(2)2| + c [since ∫ 1/√x2-a2 dx =log|x +√x2-a2| + c]
= log|t-1 +√(t-1)2-4| + c
= log|t-1 +√t2-2t+1-4| + c
= log|t-1 +√t2-2t-3| + c
= log|sinx-1 +√sin2x-2sinx-3| + c
Hence, I = log|sinx-1 +√sin2x-2sinx-3| + c
问题16.评估∫√cosecx-1dx
解决方案:
Let us assume I =∫ √cosecx-1 dx
= ∫ √1/sinx -1 dx
=∫ √1-sinx /sinx dx
=∫ √(1-sinx)+(1+sinx) /(1+sinx)sinx dx
=∫ √(1+sinx-sinx-sin2x) /sin2x+sinx dx
=∫ √cos2x /sin2x+sinx dx
=∫ cosx /√sin2x+sinx dx (i)
Let sinx = t
cosx dx = dt
Put the above value in eq. (i)
= ∫ dt/√t2+t
= ∫ dt/√t2+2t(1/2)+(1/2)2-(1/2)2
= ∫ dt/√(t+1/2)2-(1/2)2 (ii)
Let t+1/2 = u
dt = du
Put the above value in eq. (ii)
= ∫ dt/√(u)2-(1/2)2
Integrate the above eq. then, we get
= log|u +√u2-(1/2)2| + c [since ∫ 1/√x2-a2 dx =log|x +√x2-a2| + c]
= log|t+1/2 +√(t+1/2)2-(1/2)2| + c
= log|t+1/2 +√t2+t| + c
Hence, I = log|sinx+1/2 +√sin2x+sinx| + c
问题17.评估∫sinx-cosx /√sin2xdx
解决方案:
Let us assume I =∫ sinx-cosx/ √sin2x dx
∫ sinx-cosx/ √sin2x dx = ∫ sinx-cosx/ √(sinx+cosx)2 -1 dx
= ∫ sinx-cosx/ √(sinx+cosx)2 -1 dx (i)
Let sinx+cosx = t
cosx-sinx dx = dt
Put the above value in eq. (i)
= -∫ dt/ √t2-(1)2
Integrate the above eq. then, we get
= – log|t +√t2-(1)2| + c [since ∫ 1/√x2-a2 dx =log|x +√x2-a2| + c]
= – log|sinx+cosx +√sin2x| + c
Hence, I = – log|sinx+cosx +√sin2x| + c