Let f(x)=cosxsqrt(1+sinx).
A. Let F(x)=the integral of f(x)dx and F(0)=5/3, find F(pi/2).
I got 3.386 with U-substitution
B. If G(x)=the integral of f(x)sinxdx and G(0)=1/3, find G(x).
I got a reply to this but I did not understand the explanation. How would I solve this with U-substitution.
Let G=integral cosxsinxsqrt(1+sinx)dx let u=1+sinx du/dx=cosx dx=du/cosx substituting dx cancel cos x in the integral so It become:integral(u-1)sqrtudu(since sinx=u-1) therefore open bracket gives:u^1/2(u-1)du=u^3/2-u^1/2du integrate each gives:2/5u^5/2-2/3u^3/2 substitute u gives 2/5(1+sinx)^5/2-2/3(1+sinx)^3/2+c
Sorry about my half-angle formula hint. Got me nowhere. Instead, for F(x), let
u^2 = 1+sinx
2u du = cosx dx
and now you have
∫2u^2 du = 2/3 u^3 = 2/3 (1+sinx)^(3/2) + c
Now you can find c, since you know F(0)
To solve these integrals using u-substitution, we need to choose an appropriate substitution that simplifies the integrand. Let's go through each problem step-by-step.
A. Let F(x) = ∫f(x)dx, where f(x) = cos(x)√(1+sin(x)). Given F(0) = 5/3, we want to find F(pi/2).
1. Choose the substitution:
Let u = 1 + sin(x). Then, du/dx = cos(x) and dx = du/cos(x).
2. Rewrite the integral in terms of u:
F(x) = ∫f(x)dx = ∫cos(x)√(1+sin(x))dx
= ∫√u (du/cos(x))
= ∫√u du
= (√u^3)/3 + C, where C is the constant of integration.
3. Substitute the limits of integration:
Applying the limits F(0) = 5/3, we have:
(√(1+sin(0))^3)/3 + C = 5/3
(√1^3)/3 + C = 5/3
(1/3) + C = 5/3
C = 4/3
4. Evaluate F(pi/2):
F(pi/2) = (√(1+sin(pi/2))^3)/3 + C
= (√(1+1)^3)/3 + (4/3)
= (√2^3)/3 + 4/3
= (2√2)/3 + 4/3
= (2√2 + 4)/3
Therefore, F(pi/2) = (2√2 + 4)/3 ≈ 3.386.
B. Let G(x) = ∫f(x)sin(x)dx, where f(x) = cos(x)√(1+sin(x)). Given G(0) = 1/3, we want to find G(x).
1. Choose the substitution:
Let u = 1 + sin(x). Then, du/dx = cos(x) and dx = du/cos(x).
2. Rewrite the integral in terms of u:
G(x) = ∫f(x)sin(x)dx
= ∫cos(x)√(1+sin(x))sin(x)dx
= ∫√u (du/cos(x))sin(x)dx
= ∫√u sin(x) du
= ∫√u (-d(cos(x))) du
= -∫√u d(cos(x)).
Note that we used the fact that d(cos(x)) = -sin(x)dx.
3. Integrate G(x) with respect to u:
G(x) = -∫√u d(cos(x))
= -∫u^0.5 d(cos(x))
= -2/3u^1.5 + C, where C is the constant of integration.
4. Substitute the limits of integration:
Applying the limits G(0) = 1/3, we have:
-2/3(0.5)u^1.5 + C = 1/3
-1/3u^1.5 + C = 1/3
C = 2/3
5. Simplify G(x) using the original variable x:
G(x) = -2/3(1 + sin(x))^1.5 + 2/3
Therefore, G(x) = -2/3(1 + sin(x))^1.5 + 2/3.