Find the integral of sinh^2(x)dx.
just like regular trig functions,
sinh^2(x) = (cosh(2x)-1)/2
∫(cosh(2x)-1)/2 = 1/2 (1/2 sinh(2x) - x)
To find the integral of sinh^2(x), we can use the identity: sinh^2(x) = (cosh(2x) - 1)/2.
Let's denote the integral of sinh^2(x) as I. Then we have:
I = ∫sinh^2(x)dx
= ∫[(cosh(2x) - 1)/2]dx
= ∫cosh(2x)dx/2 - ∫dx/2.
Now, let's find the indefinite integrals of each term separately:
∫cosh(2x)dx/2 = (1/2)∫cosh(2x)dx.
To integrate cosh(2x), we can use the substitution method. Let u = 2x, then du = 2dx. Rearranging, we have dx = du/2.
Now substitute u = 2x and dx = du/2 into the integral:
(1/2)∫cosh(2x)dx = (1/2)∫cosh(u) * (du/2)
= (1/4) ∫cosh(u)du
= (1/4)sinh(u) + C,
where C is the constant of integration.
Since u = 2x, we have:
(1/4)sinh(u) + C = (1/4)sinh(2x) + C.
Now let's continue with the second term:
∫dx/2 = (1/2)∫dx.
Integrating dx, we get:
(1/2)∫dx = (1/2)x + C,
where C is the constant of integration.
Putting it all together, we get the final result:
I = (1/4)sinh(2x) + (1/2)x + C.
So, the integral of sinh^2(x) is (1/4)sinh(2x) + (1/2)x + C.
To find the integral of sinh^2(x)dx, you can use integration by substitution or integration by parts.
Let's use integration by substitution:
Step 1: Start by letting u = sinh(x).
Taking the derivative of u with respect to x gives du/dx = cosh(x).
Step 2: Rearrange the equation to solve for dx: dx = du / cosh(x).
Step 3: Rewrite the integral using the substitution: ∫sinh^2(x)dx = ∫u^2 (du / cosh(x)).
Step 4: Substitute u^2 and du / cosh(x) into the integral:
∫(u^2 / cosh(x))du.
Step 5: Rewrite cosh(x) in terms of u using the identity cosh^2(x) = sinh^2(x) + 1:
cosh(x) = sqrt(sinh^2(x) + 1).
Therefore, cosh(x) = sqrt(u^2 + 1).
Step 6: Substitute sqrt(u^2 + 1) in place of cosh(x) in the integral:
∫(u^2 / sqrt(u^2 + 1))du.
Step 7: Now, you can integrate this expression using a simple power rule:
∫(u^2 / sqrt(u^2 + 1))du = (2/3)(u^3)(sqrt(u^2 + 1)) + C,
where C is the constant of integration.
Step 8: Substitute u back into the equation in terms of sinh(x):
(2/3)(sinh(x)^3)(sqrt(sinh(x)^2 + 1)) + C.
Therefore, the integral of sinh^2(x)dx is (2/3)(sinh(x)^3)(sqrt(sinh(x)^2 + 1)) + C.