Session 73 Worked Sample: http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/unit-4-techniques-of-integration/part-a-trigonometric-powers-trigonometric-substitution-and-completing-the-square/session-73-completing-the-square/MIT18_01SCF10_ex73prb.pdf I substituted x-1 with cosu, and the result came out as arccos(x-1). But the online solution is arcsin(x-1) because it substitutes x-1 with sinx. These results are not the same, are they? Why?

Question

Session 73 Worked Sample: http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/unit-4-techniques-of-integration/part-a-trigonometric-powers-trigonometric-substitution-and-completing-the-square/session-73-completing-the-square/MIT18_01SCF10_ex73prb.pdf   I substituted x-1 with cosu, and the result came out as arccos(x-1). But the online solution is arcsin(x-1) because it substitutes x-1 with sinx. These results are not the same, are they? Why?

anonymous · Answer

I believe you lost a minus sign somewhere along the way, and should have ended up with 
-arccos(x-1). Still not the same as arcsin(x-1), but now you have two results that differ by a constant, which is perfectly okay and not unusual when tackling an integrals involving trig functions in two different ways.