Hello younes and welcome to mathyards
<html><applet archive = "formula.jar" codebase = "." code = "formula.FormulaViewApplet.class" name = "FormulaViewApplet" width = "105" height = "28" hspace = "0" vspace = "0" align = "middle"><param name = "FORMULA_CODE" value = "INT(SUP(sin;2)xSUP(cos;2)x) dx"></applet></html>
As you may know :
sin2x = 2sinx cosx
that is :
<html><applet archive = "formula.jar" codebase = "." code = "formula.FormulaViewApplet.class" name = "FormulaViewApplet" width = "101" height = "36" hspace = "0" vspace = "0" align = "middle"><param name = "FORMULA_CODE" value = "sinx cosx = DIV(1;2)sin2x"></applet></html>
Squaring both sides :
<html><applet archive = "formula.jar" codebase = "." code = "formula.FormulaViewApplet.class" name = "FormulaViewApplet" width = "144" height = "36" hspace = "0" vspace = "0" align = "middle"><param name = "FORMULA_CODE" value = "SUP(sin;2)xSUP(cos;2)x = DIV(1;2)SUP(sin;2)2x"></applet></html>
The integral becomes:
<html><applet archive = "formula.jar" codebase = "." code = "formula.FormulaViewApplet.class" name = "FormulaViewApplet" width = "144" height = "36" hspace = "0" vspace = "0" align = "middle"><param name = "FORMULA_CODE" value = "SUP(sin;2)xSUP(cos;2)x = DIV(1;2)SUP(sin;2)2x"></applet></html>
Now:
<html><applet archive = "formula.jar" codebase = "." code = "formula.FormulaViewApplet.class" name = "FormulaViewApplet" width = "102" height = "36" hspace = "0" vspace = "0" align = "middle"><param name = "FORMULA_CODE" value = "SUP(sin;2)2x = DIV(1- cos4x;2)"></applet></html>
Substituting in the integral
<html><applet archive = "formula.jar" codebase = "." code = "formula.FormulaViewApplet.class" name = "FormulaViewApplet" width = "88" height = "36" hspace = "0" vspace = "0" align = "middle"><param name = "FORMULA_CODE" value = "DIV(1;4)INT(DIV(1- cos4x;2) dx)"></applet></html>
<html><applet archive = "formula.jar" codebase = "." code = "formula.FormulaViewApplet.class" name = "FormulaViewApplet" width = "98" height = "36" hspace = "0" vspace = "0" align = "middle"><param name = "FORMULA_CODE" value = "DIV(1;8)INT((1- cos4x)) dx"></applet></html>
with
<html><applet archive = "formula.jar" codebase = "." code = "formula.FormulaViewApplet.class" name = "FormulaViewApplet" width = "112" height = "36" hspace = "0" vspace = "0" align = "middle"><param name = "FORMULA_CODE" value = "INT(cos4x) dx = DIV(1;4)sin4x"></applet></html>
The final answer:
<html><applet archive = "formula.jar" codebase = "." code = "formula.FormulaViewApplet.class" name = "FormulaViewApplet" width = "94" height = "36" hspace = "0" vspace = "0" align = "middle"><param name = "FORMULA_CODE" value = "DIV(x;8) - DIV(1;32)sin4x + C"></applet></html>