 Hello students, let's work out the following problem. It says evaluate the integral cos x to the power 4 dx. So let's now start the solution and let I be the integral cos x to the power 4 dx and this can be written as cos square x to the power 2 dx. Now we know that 2 cos square theta is equal to 1 plus cos 2 theta. So this implies cos square theta is equal to 1 plus cos 2 theta by 2. So here we'll make the substitution. So we have cos square x as 1 plus cos 2x upon 2 whole square dx. Now again, we have 1 by 4 into integral. Here we'll apply the formula of a plus b whole square. So we have 1 square that is 1 plus cos square 2x plus 2 into 1 into cos 2x is 2 cos 2x dx. Now again, this is equal to 1 by 4 into integral 1 plus again, we'll use the same formula which we used above. So this becomes 1 plus cos 2 into 2x upon 2 plus 2 cos 2x dx. This is again equal to 1 by 4 integral. 1 plus 1 plus cos 4x upon 2 plus 2 cos 2x dx. This is again equal to 1 by 4 into integral 1 plus 1 by 2 plus cos 4x by 2 plus 2 cos 2x dx. 1 by 4 integral 1 plus 1 by 2 is 3 by 2 plus cos 4x by 2 plus 2 into cos 2x dx. Now the integral of 3 by 2 with respect to x is 3 by 2x. So we have 1 by 4 into 3 by 2x plus 1 by 2 into integral of cos 4x is sin 4x upon 4 plus 2 into integral of cos 2x is sin 2x upon 2 plus c where c is the constant of integration. So here we have 1 by 4 into 3 by 2x plus 1 by 8 into sin 4x plus sin 2x plus c. Now again this is equal to 1 by 4 into taking lcm, lcm would be 8 and here we'll have 12x plus sin 4x plus 8 sin 2x plus c and here we'll have 8 in the denominator. So this is equal to 1 by 32 into 12x plus 8 sin 2x plus sin 4x plus c and this is the required value of the integral. So this completes the question and the session. Bye for now. Take care. Have a good day.