Question 11:
Given ∫3−2g(x)dx=4, and ∫3−2h(x)dx=9, find the value of(a) ∫3−25g(x)dx,(b) m if ∫3−2[g(x)+3h(x)+4m]dx=12
Solution:
(a)
∫3−25g(x)dx=5∫3−2g(x)dx =5×4 =20
(b)
∫3−2[g(x)+3h(x)+4m]dx=12∫3−2g(x)dx+3∫3−2h(x)dx+∫3−24mdx=124+3(9)+4m[x]3−2=12 4m[3−(−2)]=−19 20m=−19 m=−1920
Given ∫3−2g(x)dx=4, and ∫3−2h(x)dx=9, find the value of(a) ∫3−25g(x)dx,(b) m if ∫3−2[g(x)+3h(x)+4m]dx=12
Solution:
(a)
∫3−25g(x)dx=5∫3−2g(x)dx =5×4 =20
(b)
∫3−2[g(x)+3h(x)+4m]dx=12∫3−2g(x)dx+3∫3−2h(x)dx+∫3−24mdx=124+3(9)+4m[x]3−2=12 4m[3−(−2)]=−19 20m=−19 m=−1920
Question 12:
(a) Find the value of ∫1−1(3x+1)3dx.(b) Evaluate ∫431√2x−4 dx.
Solution:
a) ∫1−1(3x+1)3dx=[(3x+1)44(3)]1−1 =[(3x+1)412]1−1 =112[44−(−2)4] =112(256−16) =20
(b) ∫431√2x−4 dx=∫431(2x−4)12 dx =∫43(2x−4)−12 dx =[(2x−4)−12+112(2)]43 =[√2x−4]43 =[√2(4)−4−√2(3)−4] =2−√2
(a) Find the value of ∫1−1(3x+1)3dx.(b) Evaluate ∫431√2x−4 dx.
Solution:
a) ∫1−1(3x+1)3dx=[(3x+1)44(3)]1−1 =[(3x+1)412]1−1 =112[44−(−2)4] =112(256−16) =20
(b) ∫431√2x−4 dx=∫431(2x−4)12 dx =∫43(2x−4)−12 dx =[(2x−4)−12+112(2)]43 =[√2x−4]43 =[√2(4)−4−√2(3)−4] =2−√2
Question 13:
Given that y=x22x−1, show thatdydx=2x(x−1)(2x−1)2. Hence, evaluate ∫2−2x(x−1)4(2x−1)2 dx.
Solution:
y=x22x−1dydx=(2x−1)(2x)−x(2)(2x−1)2 =4x2−2x−2x2(2x−1)2 =2x2−2x(2x−1)2 =2x(x−1)(2x−1)2 (shown)∫2−22x(x−1)(2x−1)2 dx=[x22x−1]2−218∫2−22x(x−1)(2x−1)2 dx=18[x22x−1]2−214∫2−2x(x−1)(2x−1)2 dx=18[(222(2)−1)−((−2)22(−2)−1)] =18[(43)−(4−5)] =18(3215) =415
Given that y=x22x−1, show thatdydx=2x(x−1)(2x−1)2. Hence, evaluate ∫2−2x(x−1)4(2x−1)2 dx.
Solution:
y=x22x−1dydx=(2x−1)(2x)−x(2)(2x−1)2 =4x2−2x−2x2(2x−1)2 =2x2−2x(2x−1)2 =2x(x−1)(2x−1)2 (shown)∫2−22x(x−1)(2x−1)2 dx=[x22x−1]2−218∫2−22x(x−1)(2x−1)2 dx=18[x22x−1]2−214∫2−2x(x−1)(2x−1)2 dx=18[(222(2)−1)−((−2)22(−2)−1)] =18[(43)−(4−5)] =18(3215) =415