we want to find the anti-derivative or evaluate the indepth integral let's begin by factoring out the nine and write this as * integral of x 5 cine 2ar x 6 DX and now because we have x to the 6 here and x to the 5th here we'll perform U substitution and we'll let U equal x to the 6 and therefore differential U is equal to 6 x 5th DX and since we have X the 5th DX let's divide both sides by six and therefore one6 differential U is equal to X 5th DX and now let's write this in terms of U so we'll have 9 * the integral of x 5th DX again is equal to 16 duu so we'll factor out the 16 and we have differential U and x 6 is equal to U so we have the integral of cosine s u now we've made some progress here but we don't have an integration formula for cosine s u so we'll have to apply the power reducing formula given here below where if we have cosine squ U that's equal to 12 * the quantity 1 + cosine 2 U so we have 96 here which is three Hales times the integral of when we perform this substitution notice how we'll have a factor of 1/2 which we'll factor out so we have 3es * 1/2 * integral of the quantity 1 + cosine 2 U and now we'll find the anti-derivative with respect to U so here we have 34s the anti-derivative of one with respect to U would just be U and we need to be careful when integrating cosine 2 U this would require a substitution we've already used U so if we use V and let V equal 2 U notice that DV is equal to 2 du and therefore notice that du equals 12 DV so we'd have an extra Factor of2 when integrating so we'll have plus 12 * the integral of cine V which would be sin V and we know V is equal to 2 U so we have 12 sin 2 U and then plus C so writing this in terms of X we would have 34s * U which is x 6 Plus here we have 3/4 * 1 12 that would be 38 sin 2 U which is 2 x 6 this would be our anti-derivative I hope you found this helpful