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∫ (х^3/4+3√x+(5x+1) cos 2x) dx

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To integrate the given function, we need to break it down into its individual terms and integrate each term separately.

∫ (x^(3/4) + 3√x + (5x+1)cos(2x)) dx

Let's start with the first term: ∫ x^(3/4) dx

To integrate x^(3/4), we add 1 to the exponent and divide by the new exponent:

∫ x^(3/4) dx = (4/7) x^(7/4) + C

Next, let's integrate the second term: ∫ 3√x dx

To integrate √x, we add 1 to the exponent and divide by the new exponent:

∫ 3√x dx = 2(3/2) x^(3/2) + C = 3x^(3/2) + C

Now, let's integrate the third term: ∫ (5x+1)cos(2x) dx

To integrate (5x+1)cos(2x), we use integration by parts.

Let u = (5x+1) and dv = cos(2x) dx
Then du = 5 dx and v = (1/2)sin(2x)

Using the integration by parts formula: ∫ u dv = uv - ∫ v du

∫ (5x+1)cos(2x) dx = (5x+1)(1/2)sin(2x) - ∫ (1/2)sin(2x) (5 dx)

Simplifying, we have:

∫ (5x+1)cos(2x) dx = (5x+1)(1/2)sin(2x) - (5/2) ∫ sin(2x) dx

Using the identity ∫ sin(ax) dx = -1/a cos(ax), we have:

∫ (5x+1)cos(2x) dx = (5x+1)(1/2)sin(2x) - (5/2) (-1/2) cos(2x) + C

Simplifying further, we get:

∫ (5x+1)cos(2x) dx = (5x+1)(1/2)sin(2x) + (5/4) cos(2x) + C

Now, let's put everything together:

∫ (x^(3/4) + 3√x + (5x+1)cos(2x)) dx = (4/7) x^(7/4) + 3x^(3/2) + (5x+1)(1/2)sin(2x) + (5/4) cos(2x) + C

So, the final result of the integration is:

(4/7) x^(7/4) + 3x^(3/2) + (5x+1)(1/2)sin(2x) + (5/4) cos(2x) + C, where C is the constant of integration.