integral r(r^2 - 1) / (r^2+1)
integral (e^t-e^-t)^2
integral (1-2x)^2 / x
To solve these integrals, we will use different techniques depending on the complexity of the integrand. Here's how to solve each of them step by step:
1) Integral of (r(r^2 - 1)) / (r^2 + 1):
To solve this integral, we can perform polynomial long division to simplify the integrand. Divide r^3 - r by r^2 + 1:
(r^3 - r) / (r^2 + 1) = r - 1 + (1 / (r^2 + 1))
Now we can rewrite the original integral as:
∫ [r - 1 + (1 / (r^2 + 1))] dr
The first term r is straightforward to integrate: (1/2)r^2. The second term (1 / (r^2 + 1)) can be rewritten as 1 / ((r^2 + 1)^{1/2}) and then integrated as inverse hyperbolic tangent: arctanh(r). The final integral becomes:
∫ (1/2)r^2 - ∫ arctanh(r) dr + C,
where C is the constant of integration.
2) Integral of (e^t - e^(-t))^2:
To solve this integral, expand the square and simplify the integrand:
(e^t - e^(-t))^2 = (e^t)^2 - 2(e^t)(e^(-t)) + (e^(-t))^2
= e^(2t) - 2 + e^(-2t)
Now we can split the integral into three separate integrals:
∫ e^(2t) dt - ∫ 2 dt + ∫ e^(-2t) dt
Each of the three integrals is easy to evaluate:
∫ e^(2t) dt = (1/2)e^(2t)
∫ 2 dt = 2t
∫ e^(-2t) dt = (-1/2)e^(-2t)
Adding them up, we get:
∫ (e^t - e^(-t))^2 dt = (1/2)e^(2t) - 2t - (1/2)e^(-2t) + C,
where C is the constant of integration.
3) Integral of (1-2x)^2 / x:
To solve this integral, expand and simplify the integrand:
(1-2x)^2 / x = (1 - 4x + 4x^2) / x
= 1/x - 4 + 4x
Now we can rewrite the integral as:
∫ (1/x - 4 + 4x) dx
The first term 1/x can be integrated as ln|x|, the second term -4 is a constant, and the last term 4x can be integrated as (2x^2). The final integral becomes:
∫ 1/x dx - ∫ 4 dx + ∫ 4x dx
= ln|x| - 4x + 2x^2 + C,
where C is the constant of integration.