第12类RD Sharma解–第19章不定积分–练习19.32

问题1.∫1/ [(x − 1)√(x + 2)] dx

解决方案：

We have,

∫1/[(x − 1)√(x + 2)]dx

Let x + 2 = t², so we get, xdx = 2tdt

So, the equation becomes,

= ∫2t/(t² − 3)(t)dt

= 2∫dt/(t² − 3)

= (2/2√3) log |(t − √3)/(t + √3)| + c

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

问题2。∫1/ [(x − 1)√(2x + 3)] dx

解决方案：

We have,

∫1/[(x − 1)√(2x + 3)]dx

Let 2x + 3 = t², so we have, 2dx = 2tdt,

=> dx = tdt

So, the equation becomes,

= ∫t/[(t² − 3 − 2)/2](t)dt

= 2∫dt/(t² − 5)

= (2/2√5) log |(t − √5)/(t + √5)| + c

= (1/√5) log |(√(2x + 3) − √5)/(√(2x + 3) + √5)| + c 

问题3.∫(x + 1)/ [(x − 1)√(x + 2)] dx

解决方案：

We have,

∫(x + 1)/[(x − 1)√(x + 2)]dx

= ∫(x − 1 + 2)/[(x − 1)√(x + 2)]dx

= ∫(x − 1)/[(x − 1)√(x + 2)]dx + ∫2/[(x − 1)√(x + 2)]dx

= ∫(dx/√(x + 2)] + 2∫dx/[(x − 1)√(x + 2)]

In second part, let x + 2 = t², so we get, xdx = 2tdt

So, the equation becomes,

= ∫(dx/√(x + 2)] + ∫2t/(t² − 3)(t)dt

= ∫(dx/√(x + 2)] + 2∫dt/(t² − 3)

= 2√(x + 2) + c₁ + (4/2√3) log |(t − √3)/(t + √3)| + c₂

= 2√(x + 2) + (2/√3) log |(√(x − 2) − √3)/(√(x − 2)+√3)| + c

问题^4∫x2 / [(X – 1)√(X + 2)] DX

解决方案：

We have,

∫x²/[(x − 1)√(x + 2)]dx 

= ∫(x² − 1 + 1)/[(x − 1)√(x + 2)]dx 

= ∫(x − 1)(x + 1)/[(x − 1)√(x + 2)]dx + ∫dx/[(x − 1)√(x + 2)]

= ∫(x + 1)/[√(x + 2)]dx + ∫dx/[(x − 1)√(x + 2)]

= ∫[(x + 2) − 1]/[√(x + 2)]dx + ∫dx/[(x − 1)√(x + 2)]

= ∫√(x + 2)dx − ∫dx/[√(x + 2)] + ∫dx/[(x − 1)√(x + 2)]

In third part, let x + 2 = t², so we get, xdx = 2tdt

So, the equation becomes,

= ∫√(x + 2)dx − ∫dx/[√(x + 2)] + ∫2t/(t² − 3)(t)dt

= ∫√(x + 2)dx − ∫dx/[√(x + 2)] + 2∫dt/(t² − 3)

= (2/3)(x + 2)^3/2 + c₁ − 2√(x + 2) + c₂ + (2/2√3) log |(t − √3)/(t + √3)| + c₃

= (2/3)(x + 2)^3/2 − 2√(x + 2) + (1/√3) log |(√(x − 2) − √3)/(√(x − 2) + √3)| + c

问题5.∫x/ [(x − 3)√(x + 1)] dx

解决方案：

We have,

∫x/[(x − 3)√(x + 1)]dx

= ∫[(x − 3) + 3]/[(x − 3)√(x + 1)]dx

= ∫dx/[√(x + 1)] + 3∫dx/[(x − 3)√(x + 1)]

For second part, let x + 1 = t², so we get, dx = 2tdt.

So, the equation becomes,

= ∫dx/[√(x + 1)] + 3∫2tdt/[(t² − 4)(t)]

= 2√(x + 1) + c₁ + (3/2) log |(t − 2)/(t + 2)| + c₂

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

问题6.∫1/ [(x ² +1)√x] dx

解决方案：

We have,

∫1/[(x² + 1)√x]dx

Let x = t², so we have, dx = 2tdt

So, the equation becomes,

= 2∫t/[(t⁴ + 1)(t)]dt

= 2∫dt/(t⁴ + 1)

= 2∫(t/t²)/(t² + 1/t²)dt

= ∫[1 + 1/t² − (1 − 1/t²)]/(t² + 1/t²)dt

= ∫(1 + 1/t²)/[(t − 1/t)² + 2]dt − ∫(1 − 1/t²)/[(t + 1/t)² − 2]dt 

Let t − 1/t = y, so we have, (1 + 1/t²)dt = dy

Let t + 1/t = z, so we have, (1 − 1/t²)dt = dz

So, the equation becomes,

= ∫dy/(y² +2) − ∫dz/(z² − 2)

= (1/√2) tan⁻¹(y/√2) − (1/2√2) log |(z − √2)/(z + √2)| + c

= (1/√2) tan⁻¹[(t²−1)/√2t] − (1/2√2) log |[x + 1 − √(2x)]/[x + 1 + √(2x)]| + c

= (1/√2) tan⁻¹[(x−1)/√(2x)] − (1/2√2) log |[x + 1 − √(2x)]/[x + 1 + √(2x)]| + c

问题7.∫x/ [(x ² + 2x + 2)√(x + 1)] dx

解决方案：

We have,

∫x/[(x² + 2x + 2)√(x + 1)]dx

Let x + 1 = t², so we have, dx = 2tdt

So, the equation becomes,

= 2∫(t² − 1)(t)/[(t⁴ + 1)(t)]dt

= 2∫(t² − 1)/(t⁴ + 1)dt

= 2∫(1 − 1/t²)/[(t + 1/t)² − 2]dt

Let t + 1/t = y, so we have, (1 − 1/t²)dt = dy

So, the equation becomes,

= 2∫dy/(y² − 2)

= (2/2√2) log |(y − √2)/(y + √2)| + c

= (1/√2) log |(t² + 1 − √2t)/(t² + 1 + √2t)| + c

= (1/√2) log |[x + 2 − √(2x + 2)]/[x + 2 + √(2x + 2)]| + c

问题8.∫1/ [(x − 1)√(x ² +1)] dx

解决方案：

We have,

∫1/[(x − 1)√(x² + 1)]dx

Let x − 1 = 1/t, so we have dx = (−1/t²)dt

So, the equation becomes,

= −∫(1/t²)/[(1/t)√[(1 + 1/t)² + 1]]dt

= −∫dt/√(2t² + 2t + 1)

= −(1/√2)∫dt/√(t² + t + 1/2)

= −(1/√2)∫dt/√[(t + 1/2)² + 1/4]

= −(1/√2) log |(t + 1/2) + √[(t + 1/2)² + 1/4]| + c

= −(1/√2) log |(1/(x − 1) + 1/2) + √[(1/(x−1) + 1/2)² + 1/4]| + c

问题9.∫1/ [(x + 1)√(x ² + x + 1)] dx

解决方案：

We have,

∫1/[(x + 1)√(x² + x + 1)]dx

Let x + 1 = 1/t, so we have dx = (−1/t²)dt

So, the equation becomes,

= −∫(1/t²)/[(1/t)√(1/t² + 1/t − 1)]dt

= −∫dt/√(1 + t − t²)

= −∫dt/√[5/4 − (1/4 − t + t²)]

= −∫dt/√[5/4 − (t − 1/2)²]

= − sin⁻¹[(t − 1/2)/(√5/2)] + c

= − sin⁻¹[(2t − 1)/√5] + c

= − sin⁻¹[(1 − x)/[√5(x + 1)]] + c

问题10.∫1/ [(x ² − 1)√(x ² +1)] dx

解决方案：

We have,

∫1/[(x² − 1)√(x² + 1)]dx

Let x = 1/t, so we get, dx = (−1/t²)dt

So, the equation becomes,

= −∫(1/t²)/[(1/t² − 1)√(1/t² + 1)]dt

= −∫t/[(1 − t²)√(1 + t²)]dt

Let 1 + t² = y², so we have, 2tdt = 2ydy

=> tdt = ydy

So, the equation becomes,

= ∫ydy/(y² − 2)y

= ∫dy/(y² − 2)

= (1/2√2) log |(y − √2)/(y + √2)| + c

= (1/2√2) log |(y − √2)/(y + √2)| + c

= (1/2√2) log |[√(1 + t²) − √2]/[√(1 + t²) + √2]| + c

= −(1/2√2) log |[√2x + √(x² + 1)]/[√2x − √(x² + 1)]| + c

问题11.∫x/ [(x ² + 4)√(x ² +1)] dx

解决方案：

We have,

∫x/[(x² + 4)√(x² + 1)]dx

Let x² + 1 = t², so we get, 2xdx = 2tdt

=> xdx = tdt

So, the equation becomes,

= ∫t/(t² + 3)(t)dt

= ∫dt/(t² + 3)

= (1/√3) tan⁻¹(t/√3) + c

= (1/√3) tan⁻¹[√(x² + 1)/√3] + c

问题12.∫1/ [(1 + x ² )√(1 − x ² )] dx

解决方案：

We have,

∫1/[(1 + x²)√(1 − x²)]dx 

Let x = 1/t, so we get, dx = (−1/t²)dt

So, the equation becomes,

= −∫(1/t²)/[(1/t² + 1)√(1 − 1/t²)]dt

= −∫t/[(t² + 1)√(t² − 1)]dt

Let t² − 1 = y², so we get, 2tdt = 2ydy

=> tdt = ydy

So, the equation becomes,

= −∫y/[(y² + 2)(y)]dy

= −∫1/(y² + 2)dy

= −(1/√2) tan⁻¹(y/√2) + c

= −(1/√2) tan⁻¹(√(t² − 1)/√2) + c

= −(1/√2) tan⁻¹(√(1 − x²)/√2x) + c

问题13.∫1/ [(2x ² + 3)√(x ² − 4)] dx

解决方案：

We have,

∫1/[(2x² + 3)√(x² − 4)]dx 

Let x = 1/t, so we have dx = (−1/t²)dt

So, the equation becomes,

= −∫(1/t²)/[(2/t² + 3)√(1/t² − 4)]dt

= −∫t/[(2 + 3t²)√(1−4t²)]dt

Let 1 − 4t² = y², so we get, −8tdt = 2ydy

So, the equation becomes,

= (1/4) ∫y/[(11 − 3y²)y/4]dy

= (1/3) ∫1/(11/3 − y²)dy

= (1/2√33) log |[y − √(11/3)]/[y + √(11/3)]| + c

= (1/2√33) log |[√(1 − 4t²) − √(11/3)]/[√(1 + 4t²) + √(11/3)]| + c

= (1/2√33) log |[√(11x) + √(3x² − 12)]/[√(11x) − √(3x² − 12)]| + c

问题14.∫x/ [(x ² + 4)√(x ² + 9)] dx

解决方案：

We have,

∫x/[(x² + 4)√(x² + 9)]dx 

Let x² + 9 = y², so we have 2xdx = 2ydy

=> xdx = ydy

So, the equation becomes,

= ∫y/[(y² − 5)y]dy

= ∫1/(y² − 5)dy

= (1/2√5) log |(y − √5)/(y + √5)| + c

= (1/2√5) log |(√(x² + 9) − √5)/(√(x² + 9) + √5)| + c