Integrate ( x^2cosx)/(1+sinx)^2 dx
with limits from 0 to pi/4
To solve the integral ∫(x^2*cosx)/(1+sinx)^2 dx with limits from 0 to π/4, we will use integration by parts.
Let's start by identifying the parts of the integrand that we can assign as "u" and "dv."
Let u = x^2 and dv = (cosx)/(1+sinx)^2 dx.
Next, we will find du and v by differentiating u and integrating dv, respectively.
Differentiate u: du = 2x dx
Integrate dv:
To integrate the integral of (cosx)/(1+sinx)^2 dx, we will use a substitution.
Let t = 1 + sinx, then dt = cosx dx.
The integral becomes:
∫(cosx)/(1+sinx)^2 dx = ∫dt/t^2
= -1/t + C
= -1/(1+sinx) + C
Now we have our values for du and v:
du = 2x dx
v = -1/(1+sinx)
Applying integration by parts formula:
∫(u dv) = uv - ∫(v du),
we get:
∫(x^2*cosx)/(1+sinx)^2 dx = (-x^2/(1+sinx)) - ∫(-1/(1+sinx) * 2x dx)
= -x^2/(1+sinx) + 2∫(x/(1+sinx) dx)
The integral on the right side, ∫(x/(1+sinx) dx, can be solved using a substitution.
Let u = 1 + sinx, then du = cosx dx.
The integral becomes:
2∫(x/(1+sinx) dx = 2∫(x/u du)
= 2∫(1 - sinx/u du)
= 2(u - ∫sinx/u du)
= 2u - 2∫sinx/u du
Now we have our new integral:
2u - 2∫sinx/u du
To evaluate this integral, we need to find the antiderivative of sinx/u.
Let's find dv and v:
dv = sinx/u dx
v = ln|u|
Using the integration by parts formula again:
∫(sinx/u) dx = - cosx/u + ∫(cosx/u^2) dx
= -cosx/u + ∫u^(-2)cosx dx
This new integral can be solved as:
∫(u^(-2)cosx) dx = u^(-2)sinx - ∫(-2u^(-3)sinx) dx
= u^(-2)sinx + 2∫(u^(-3)sinx) dx
Now we have a new integral to solve:
2∫(u^(-3)sinx) dx
Using the integration by parts formula again:
Letting w = u^(-3) and dz = sinx dx,
dw = -3u^(-4) du and z = -cosx
The integral becomes:
2∫(u^(-3)sinx) dx = -2u^(-3)cosx - 2∫(-3u^(-4)cosx) dx
= -2u^(-3)cosx + 6∫(u^(-4)cosx) dx
This new integral can be solved as:
6∫(u^(-4)cosx) dx = 6(u^(-4)sinx - ∫(-4u^(-5)sinx) dx)
= 6u^(-4)sinx + 24∫(u^(-5)sinx) dx
Using the integration by parts formula one last time:
Letting p = u^(-5) and dq = sinx dx,
dp = -5u^(-6) du and q = -cosx
The integral becomes:
24∫(u^(-5)sinx) dx = -24u^(-5)cosx - 24∫(-5u^(-6)cosx) dx
= -24u^(-5)cosx + 120∫(u^(-6)cosx) dx
Now, we have one final integral left to solve:
120∫(u^(-6)cosx) dx
Using the integration by parts formula one last time:
Letting r = u^(-6) and dm = cosx dx,
dr = -6u^(-7) du and m = sinx
The integral becomes:
120∫(u^(-6)cosx) dx = -120u^(-6)sinx + 720∫(u^(-7)sinx) dx
We can see that this new integral is similar to the previous one, so we can repeat the process until we reach an integral that we can solve.
Continuing with the process of integration by parts, we eventually arrive at an integral that simplifies to a constant term. This allows us to solve the entire expression.
However, due to the complex and lengthy nature of this integration, it is not recommended to solve it step-by-step manually. Instead, it is recommended to use computer software or calculators capable of performing symbolic integration, such as Wolfram Alpha or MATLAB.
To integrate the given function, we will use the technique of integration by substitution. Here's how we can do it step by step:
1. Let u = 1 + sin(x). Then, differentiate both sides with respect to x to find du/dx = cos(x).
2. Rearrange the equation to solve for dx: dx = du/cos(x).
3. Substitute the value of u and dx into the original expression to get the new integral:
∫ (x^2 * cos(x))/(1+sin(x))^2 dx = ∫ [(x^2 * cos(x))/(u^2)] * (du/cos(x)).
4. Simplify the expression: ∫ x^2/u^2 du.
5. Now that we have a simpler integral, we can solve it. Rewrite x^2 as x * x, and then separate the variables to get: ∫ (x/u^2) * (x/u^2) du.
6. Express the integral as the product of two separate integrals: ∫ (x/u^2) du * ∫ (x/u^2) du.
7. Integrate the first part: ∫ (x/u^2) du = ∫ (x/(1+sin(x))^2) du = ∫ (x/u^2) du = -x/u + C1, where C1 is the constant of integration for the first integral.
8. Integrate the second part: ∫ (x/u^2) du = -x/u + C2, where C2 is the constant of integration for the second integral.
9. Finally, combine the two results: -x/u + C1 * -x/u + C2 = -x/u - x/u + C1 + C2 = -2x/u + C, where C = C1 + C2.
10. Convert u back to the original variable x: -2x/(1+sin(x)) + C.
11. Evaluate the definite integral with the limits from 0 to π/4:
-2(π/4)/(1+sin(π/4)) + 2(0)/(1+sin(0)) = -π/2/(1+1/sqrt(2)) = -π/2/(sqrt(2)+1) + C.
So, the final answer to the integral is -π/2/(sqrt(2)+1) + C, where C represents the constant of integration.
but i want step by step
Wolfram gives us this
http://integrals.wolfram.com/index.jsp?expr=%28+x%5E2cosx%29%2F%281%2Bsinx%29%5E2&random=false
I have no clue how they got that.