Find an indefinite integral: I = ∫ xdx / √ [1 + x ^ 2 + √ (1 + x ^ 2) ^ 3]

Find an indefinite integral: I = ∫ xdx / √ [1 + x ^ 2 + √ (1 + x ^ 2) ^ 3]

The following is a direct simplification:
I=∫xdx/√[1+x^2+√(1+x^2)^3]
=∫xdx/√(1+x^2)√[1+√(1+x^2)]
=1/2 ∫d(1+x^2)/√(1+x^2)√[1+√(1+x^2)]
=∫d[1+√(1+x^2)]/√[1+√(1+x^2)]
=2√[1+√(1+x^2)]+C
The above solution is mainly differential, relatively speaking, it requires a lot of mathematical thinking. Another way is to open the radical and change the element so that T ^ m = 1 + x ^ 2 (the value of M depends on the number of roots of the problem)

∫ 2 ^ xdx / √ 1-4 ^ x for indefinite integral

Calculate the definite integral,
The upper limit of cos2xdx is π / 4 and the lower limit is π / 6
Why is it equal to this. 1/2*sin2x

∫ cos2xdx upper limit is π / 4 lower limit is π / 6 = 1 / 2 * sin2x upper limit is π / 4 lower limit is π / 6
=1 / 2Sin π / 2-1 / 2Sin π / 3 = 1 / 2-radical 3 / 4. I am a master of mathematics. I can't figure out mathematical problems. Get a master of mathematics!