Solve math related problems

Part of the answer refers to GPT, GPT_Pro better solve the problem

solution: Jax back(= 10] ∫ a ^ x cV(t) dt = ∫ a ^ x c(t) V '(t) dt = c(x) V(x) - ∫ a ^ x c'(t) V(t) dt

dx

solution: jJxsexdx = ∫ x ^ se ^ XDX =(- e ^ x x ^ s + s ∫ e ^ XDX)/(s + 1) =(- e ^ ^ s + s e x x ^ - ^ s e x a)/(s + 1).

cosx/(cosxrsinx) dx and jsinx/(cosxrsinx) dx

solution: Jcosx/(cosxrsinx) dx = ∫ cosx/(cosxrsinx) dx = ∫ / dx rsin(x) = 1 - ln | | rsin x + C

jsinx/(cosxrsinx) dx = ∫ sinx/(cosxrsinx) dx = 1 /(rcos x) ∫ dx = ln | | rcos x + C < br / > if you have any help answer, hope to adopt.

This answer quotes ChatGPT

$\int\frac{dx}{\sqrt{x(1+x)}}$:

We can use the formula to deform the denominator and get

where $C$is arbitrary constant.

$\ x^2 e^x dx$:

We can use integration by parts to solve this integral:

where $C$is an arbitrary constant.

three, $\ int \ frac {\ cos x} {\ cos + x \ sin x} dx $and $\ int \ frac {\ sin x} {\ cos + x \ sin x} dx $:

For the first integral, we can make the following deformation:

For the second integral, we can make a similar deformation:

Therefore, both integrals can be reduced to the form $\int dx$, i.e. :

where $C_1$and $C_2$are arbitrary constants.