Question

QUESTION 3 Evaluate the integral by using multiple substitutions. SV1 1 + sin2 (x-7) sin (x-7) cos (x-7) dx o 3 (1+ sinº x) (1 + sin? x)3/2 + c 3 O AV1 + sin?(x - 7) +C og (1 + sin? (x - 7)) 3/2 + c O (1 + cos2 (x - 7) 3/2 + c

Answer 1

$\bg_green \fn_phv \large \mathbf{ANSWER: \ \frac{1}{3}\left [ 1+sin^{2}(x-7) \right ]^{\frac{3}{2}}+C \ \ \ \ \ \ where \ C=integral \ constant}$
qolution : - Given to Compute It sin(x-7) Sin (x-7) CD (x-7) da. let X-7 =P Apply denivative on both sides with respect to a. (x-7)=de » dard x?= nix x-dy=de d dx ax, d n-1 dae لاله dx dre. n=1 an ☆ 1-0 = dp da. de=due a constant to de Şubstitute the terms Containing the boxes in Equation o Ijit Sinop Sinp colp de 2 Now again [let Itsin²p = q Apply derivative on Both sides with respect to p an dq ap. Jo ir Clit Sinop)= I + op I sinus Ə 2 df aq dp.

0 + 2 SinP d sinpa ap da dp d sinn=cosa da » 2 sinp. Colp 인 dp → de = 25inp Cop.dp 요 = Sinp Cosp de 2 2 Now Subſtitute the terms Containing the Boxes in Equation @. i.e Sjit Sintp Sipp Cosp de & fue. dq 1950년 2 1 안 을 ปี 3น ntl 2 See nt t1 x ㅋ 2 C. 2 t) 를 1 오 3 2 +C 옥 숙 2 9 +C

We know q = 1 + Sinp. put in Aloue. Desult 3 Z 2 + c » Į (+ sin på to Again we assumed p=8-7. So put in the Above result انم → 1/ 3 (it Sin? (21-7) ²+c It Sin(x-7) · Sin (x->) Cos (x-7) dx. 1 (it Sin" (x-7)+c + C, G Answer option. A