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for part a) let y=x^cos(X) , the ln(y)=ln(x^cos(X))=cos(x)ln(x), thus d/dx (ln(y(x)) = d/dx (cos(x)ln(x)), 1/y*dy/dx=cox(x)/x - sinxlnx => solve for dy/dx =>...

Integrate ln(x) by parts with u=ln(x) and dv/dx=1 Gives    x+xln(x)-x+c Therefore xln(x)+c

integral of 4x³ with respect to x =x⁴+C

3x² 6x