Tiêu đề INTEGRALS.pdf ✅

CHAPTER 7 INTEGRALS Exercise 1: NCERT Based Topic-wise MCQs 7.1 & 7.2 INTRODUCTION, INTEGRATION AS AN INVERSE PROCESS OF DIFFERENTIATION 1. ∫ x 51(tan−1 x + cot−1 x)dx NCERT Page-296/N-226 (a) x 52 52 (tan−1 x + cot−1 x) + c (b) x 52 52 (tan−1 x − cot−1 x) + c (c) πx 52 104 + π 2 + c (d) x 52 52 + π 2 + c 2. ∫ e 5log x−e 4log x e 3log x−e 2log x dx = (a) e. 3 −3x + c NCERT Page-295/N-226 (b) e 3 log x + c (c) x 3 3 + c (d) None of these 3. Evaluate : ∫ (x 2 + x) 2dx NCERT Page-296/N-229 (a) x 5 5 + x 3 3 + x 4 2 (c) 5x 5 + 3x 5 + 8x 4 (b) x 5 5 + x 3 3 + x 4 2 + c (d) 5x 5 + 3x 3 + 8x 4 + c 4. Evaluate: ∫ x 2−5x+6 x−2 dx NCERT Page-296/N-228 (a) x 2 2 − 3x (b) x − 3 (c) x 2 2 − 3x + c (d) x 2 2 + 3x + c 5. Evaluate: ∫ x 2+x+1 √x dx NCERT Page-296/N-226 (a) 5 2 x 5/2 + 3 2 x 3/2 + 1 2 x 1/2 + c (b) 2 5 x 5/2 + 3 2 x 3/2 + 2x 1/2 + c (c) x 3/2 + x 1/2 + x −1/2

(b) log tan ( x 2 − π 12) + C (c) 1 2 log tan ( x 2 + π 12) + C (d) 1 2 log tan ( x 2 − π 12) + C 13. Let g: (0, ∞) → R be a differentiable function such that ∫ ( x(cos x−sin x) e x+1 + g(x)(e x+1−xe x) (e x+1)2 ) dx = xg(x) e x+1 + c, for all x > 0, where c is an arbitrary constant. Then NCERT Page-325/N-236 (a) g is decreasing in (0, π 4 ) (b) g ′ is increasing in (0, π 4 ) (c) g + g ′ is increasing in (0, π 2 ) (d) ' g − g ′ is increasing in (0, π 2 ) 14. ∫ (x 2+1)e x (x+1) 2 dx = f(x)e x + C, where C is a constant, then d 3f dx 3 at x = 1 is equal to: (a) − 3 4 NCERT Page-326/N-236 (b) 3 4 (c) − 3 2 (d) 3 2 15. The integral ∫ (1− 1 √3 )(cos x−sin x) (1+ 2 √3 sin 2x) dx is equal to NCERT Page-304/N-238 (a) 1 2 loge | tan( x 2 + π 12) tan( x 2 + π 6 ) | + C (b) 1 2 loge | tan ( x 2 + π 6 ) tan ( x 2 + π 3 ) | + C (c) loge | tan ( x 2 + π 12) tan ( x 2 + π 12) | + C (d) 1 2 loge | tan ( x 2 − π 12) tan ( x 2 − π 6 ) | + C 16. ∫ 1 [(x−1) 3(x+2) 5]1/4 dx is equal to NCERT Page-304/N-238 (a) 4 3 ( x−1 x+2 ) 1/4 + C (b) 4 3 ( x+2 x−1 ) 1/4 + C (c) 1 3 ( x−1 x+2 ) 1/4 + C (d) 1 3 ( x+2 x−1 ) 1/4 + C 17. ∫ 2dx (e x+e−x)2 = NCERT Page-304/N-238 (a) −e −x (e x+e−x) + C (b) −1 (e x+e−x) + C
(c) 1 (e x+1)2 + C (d) 1 (e x+e−x) + C 18. If ∫ x 13/2 ⋅ (1 + x 5/2 ) 1/2 dx = A(1 + x 5/2 ) 7/2 + B(1 + x 5/2 ) 5/2 + C(1 + x 5/2 ) 3/2 + D, then (a) A = − 4 35 , B = − 8 25 , C = 4 15 (b) A = 4 35 , B = − 8 25 , C = − 4 15 (c) A = 4 35 , B = − 8 25 , C = 4 15 (d) None of these 19. ∫ (27e 9x + e 12x ) 1/3dx is equal to NCERT Page-304/N-237 (a) (1/4)(27 + e 3x) 1/3 + C (b) (1/4)(27 + e 3x) 2/3 + C (c) (1/3)(27 + e 3x ) 4/3 + C (d) (1/4)(27 + e 3x ) 4/3 + C 20. ∫ e x(1+x) cos2 (e xx) dx equals (a) −cot (ex x ) + C NCERT Page-304/N-239 (b) tan (xe x ) + C (c) tan (e x ) + C (d) cot (e x ) + C 21. Evaluate: ∫ 2 2 2 x 2 2 x 2 xdx NCERT Page-304/N-239 (a) 1 (log 2) 3 2 2 2 x + C (b) 1 (log 2) 3 2 2 x + C (c) 1 (log 2) 2 2 2 x + C (d) 1 (log 2) 4 2 2 2x + C 22. ∫ 10x 9+10x loge 10 10x+x 10 dx is equal to (a) 10x − x 10 + C NCERT Page-304/N-236 (b) 10x + x 10 + C (c) (10x − x 10) −1 + C (d) loge (logx + x 10) + C 23. ∫ cos {2tan−1 √ 1−x 1+x } dx is equal to NCERT Page-304/N-237 (a) 1 8 (x 2 − 1) + k (b) 1 2 x 2 + k (c) 1 2 x + k (d) None of these 24. ∫ e 3log x (x 4 + 1) −1dx is equal to NCERT Page-305/N-237 (a) log (x 4 + 1) + C (b) 1 4 log (x 4 + 1) + C