# Resolved questions

please help me with "easy" integrals

hi everybody,

can somebody help me with these integrals? i would be really thankful for any advice.

e^(arcsin x) dx

e^(arccos x) dx

e^(arctg x) dx

e^(arccotg x) dx

can somebody help me with these integrals? i would be really thankful for any advice.

e^(arcsin x) dx

e^(arccos x) dx

e^(arctg x) dx

e^(arccotg x) dx

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/ e^(arcsinx)dx = x * e^(arcsinx) - / x * e^(arcsinx)/sqrt(1-x^2) dx

= x * e^(arcsinx) + / 1/2 * e^(arcsinx) * 2 d sqrt(1 - x^2)

= x * e^(arcsinx) + e^(arcsinx) * sqrt(1 - x^2) - / sqrt(1 - x^2)/ sqrt(1 - x^2) e^(arcsinx) dx

= x * e^(arcsinx) + e^(arcsinx) * sqrt(1 - x^2) - / e^(arcsinx)dx

then can get: / 2e^(arcsinx)dx = x * e^(arcsinx) + e^(arcsinx) * sqrt(1 - x^2) + C'

then: / e^(arcsinx)dx = 1/2 * e^(arcsinx) [x + sqrt(1 - x^2)] + C

or : let arcsinx = u, (-pi/2 < u < pi/2 ), then x = sinu, dx = cosu du ,

/ e^(arcsinx) dx = / e^u * cos u * du

= / cos u * d e^u

= cos u * e^u + / e^u sin u du

= cos u * e^u + / sin u d e^u

= cos u * e^u + sin u * e^u - / e^u cos u du

then can get : 2 / e^u cos u du = cos u * e^u + sin u * e^u + C'

then : / e^u cos u du =1/2 [ cos u + sin u * ] * e^u + C

= 1/2 [ sqrt(1 - x^2) + x] * e^(arcsinx) + C

= x * e^(arcsinx) + / 1/2 * e^(arcsinx) * 2 d sqrt(1 - x^2)

= x * e^(arcsinx) + e^(arcsinx) * sqrt(1 - x^2) - / sqrt(1 - x^2)/ sqrt(1 - x^2) e^(arcsinx) dx

= x * e^(arcsinx) + e^(arcsinx) * sqrt(1 - x^2) - / e^(arcsinx)dx

then can get: / 2e^(arcsinx)dx = x * e^(arcsinx) + e^(arcsinx) * sqrt(1 - x^2) + C'

then: / e^(arcsinx)dx = 1/2 * e^(arcsinx) [x + sqrt(1 - x^2)] + C

or : let arcsinx = u, (-pi/2 < u < pi/2 ), then x = sinu, dx = cosu du ,

/ e^(arcsinx) dx = / e^u * cos u * du

= / cos u * d e^u

= cos u * e^u + / e^u sin u du

= cos u * e^u + / sin u d e^u

= cos u * e^u + sin u * e^u - / e^u cos u du

then can get : 2 / e^u cos u du = cos u * e^u + sin u * e^u + C'

then : / e^u cos u du =1/2 [ cos u + sin u * ] * e^u + C

= 1/2 [ sqrt(1 - x^2) + x] * e^(arcsinx) + C

= 1/2 *[ 1/2 *(-2x) /sqrt(1 - x^2)+1 + x/sqrt(1 - x^2) +1 ] * e^(arcsinx)

= 1/2 *2e^(arcsinx)

= e^(arcsinx)

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