is this answer wrong !

Yes, you could do, as there is a "c" to wrap up the constants (though you would change the cos term to a sin term as per your original question!)

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I think it must be sin4(x-3) insted of cos4(x-3) in the last step !

Yes!

Because: $$\int{\cos(4u)}\ du = \frac{1}{4}\sin(4u)$$

Also in this question, why they didn't wrote $$\frac{1}{2}(x+1)$$   insted of  $$\frac{1}{2}x$$  ! 

$$\frac{1}{2}\int (1+\cos{2(x+1)})dx=\frac{1}{2}\int dx+\frac{1}{2}\int \cos{2(x+1)}dx$$

so the single x comes from the first integral on the right-hand side. However, you could add whatever constant you like, because, as it's an indefinite integral, any other constants are accounted for in the "c" at the end.

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So here can I wrote  $$\frac{3}{8}x$$  insted of  $$\frac{3}{8}(x-3)$$   ? 

very clear

thank you Alan and heureka