Calculus question - evaluating the integral -e^cos^(-1) x dx/sqrt(1-x^2)?

Question

I came up with e^arcsin x/arcsin x + C but I am not sure I did it correctly.

Answer 1

∫ ( -e^(cosˉ¹(x)) dx / √(1 - x^2) )

First, let's write this in a much better comprehensible form.

∫ ( e^(cosˉ¹(x)) (-1/√(1 - x^2)) dx )

Here's where we have to use substitution.

Let u = cosˉ¹(x). Then
du = (-1/√(1 - x^2)) dx

{Note: (-1/√(1 - x^2)) dx is the tail end of our current integral, so it follows that du will be the tail end of our new integral.}

After the substitution, we get

∫(e^u du)

And now we integrate easily.

e^u + C

But u = cosˉ¹(x), so our final answer is

e^(cosˉ¹(x)) + C

Answer 2

? ?[a million + ?x] dx with a view to unravel this concern, evidently like we can could hire 2 categories of substitutions. enable the 1st sub. be: u = ?x du = (a million / 2?x) dx Doing a touch algebreic manipulation to the "du = " time era, we get: du = (a million / 2?x) dx (2?x)du = dx yet remember, until now we stated that u = ?x. as a result, we can substitute u in for ?x. Doing so will supply us: (2?x)du = dx (2u) du = dx in short then: u = ?x du = (a million / 2?x) dx ----------> dx = (2u) du Making those substitutions will supply us: ? 2u * ?(a million + u) du Factoring out the consistent time era: 2?u?(a million + u) du From right here, we hire the 2nd substitution. in view that we've already used a u-substitution, enable's bypass forward and hire an r substitution. Doing so will supply us: r = a million + u -----------------> u = r - a million dr = du Making those substitutions will supply us: 2?u?(a million + u) du 2?(r - a million) * ?(r) dr ² ³ Multiplying each and every little thing out will supply us: 2??(r³) - ?r dr Integrating with admire to r will supply us: (4/5)(?r?) - (4/3)*(?r³) + C 4 * [ ((?r?) / 5) - ((?r³) / 3) ] + C Substituting a million + u decrease back in for r will supply us: 4 * [ ((?(a million + u)?) / 5) - ((?(a million + u)³) / 3) ] + C Substituting ?x decrease back in for u: 4 * [ ((?(a million + ?x)?) / 5) - ((?(a million + ?x)³) / 3) ] + C very final answer: (4/5)*?((a million + ?x)?) - (4/3)*?((a million + ?x)³) ) + C

Answer 3

Let u = cos^(-1).x
du = 1/ √(1 - x²).dx
I = - ∫ e^(u) du
I = - e^(u) + c
I = - e^(cos^(-1).x) + c