Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Visit Stack Exchange
Stack Internal
Knowledge at work
Bring the best of human thought and AI automation together at your work.
Explore Stack InternalThe two evaluate to the same values.
f = 1/Sqrt[(x - t)*(y - t)]; In[5]:= v1 = Integrate[f, {t, 0, y}, Assumptions -> {0 < y < x}] Out[5]=(* Log[(x + y + 2 Sqrt[x y])/(x - y)] *) In[6]:= v2 = Integrate[f, {t, 0, x}, Assumptions -> {0 < x < y}] Out[6]=(* 2 ArcTanh[Sqrt[x/y]] *) v3 = v2 /. {y -> x, x -> y} g = v1/v3; In[23]:= Table[g /. {x -> 10}, {y, 1, 9, 1}] // N Out[23]=(* {1., 1., 1., 1., 1., 1., 1., 1., 1.} *) The two evaluate to the same values.
f = 1/Sqrt[(x - t)*(y - t)]; In[5]:= v1 = Integrate[f, {t, 0, y}, Assumptions -> {0 < y < x}] Out[5]= Log[(x + y + 2 Sqrt[x y])/(x - y)] In[6]:= v2 = Integrate[f, {t, 0, x}, Assumptions -> {0 < x < y}] Out[6]= 2 ArcTanh[Sqrt[x/y]] v3 = v2 /. {y -> x, x -> y} g = v1/v3; In[23]:= Table[g /. {x -> 10}, {y, 1, 9, 1}] // N Out[23]= {1., 1., 1., 1., 1., 1., 1., 1., 1.} The two evaluate to the same values.
f = 1/Sqrt[(x - t)*(y - t)]; v1 = Integrate[f, {t, 0, y}, Assumptions -> {0 < y < x}] (* Log[(x + y + 2 Sqrt[x y])/(x - y)] *) v2 = Integrate[f, {t, 0, x}, Assumptions -> {0 < x < y}] (* 2 ArcTanh[Sqrt[x/y]] *) v3 = v2 /. {y -> x, x -> y} g = v1/v3; Table[g /. {x -> 10}, {y, 1, 9, 1}] // N (* {1., 1., 1., 1., 1., 1., 1., 1., 1.} *) The two evaluate to the same values.
f = 1/Sqrt[(x - t)*(y - t)]; In[5]:= v1 = Integrate[f, {t, 0, y}, Assumptions -> {0 < y < x}] Out[5]= Log[(x + y + 2 Sqrt[x y])/(x - y)] In[6]:= v2 = Integrate[f, {t, 0, x}, Assumptions -> {0 < x < y}] Out[6]= 2 ArcTanh[Sqrt[x/y]] v3 = v2 /. {y -> x, x -> y} g = v1/v3; In[23]:= Table[g /. {x -> 10}, {y, 1, 9, 1}] // N Out[23]= {1., 1., 1., 1., 1., 1., 1., 1., 1.}