Find the antiderivative F of f(x)=4−3(1+x^2)^−1 that satisfies F(1)=5.
To find the antiderivative F of f(x), we need to integrate the function f(x). The integral of f(x) is denoted as ∫f(x)dx.
The function f(x) is given as:
f(x) = 4 - 3(1 + x^2)^-1
To integrate f(x), we can break it down into two parts:
∫4dx - ∫3(1 + x^2)^-1 dx
Integrating the first term is straightforward since 4 is a constant:
∫4dx = 4x + C1
Now, let's focus on the second term:
∫3(1 + x^2)^-1 dx
To integrate this term, we can use a substitution. Let's substitute u = 1 + x^2. Then, du = 2x dx, giving us dx = du / (2x).
When we substitute this into the integral, we get:
∫3/u * (du / (2x))
Next, we can simplify this expression:
∫3/(2xu) du
We still have 'x' in the denominator, which is problematic. To fix this, we can use another substitution where we substitute x^2 = t. Then, dx = (1/2) dt.
Substituting this in, we have:
∫3/(2 * √t * (1 + t)) dt
We can now integrate this expression. However, this integral is a bit more complicated and requires further techniques like partial fractions or trigonometric substitution.
Without going into too much detail, the antiderivative of the above integral can be expressed as:
F(x) = 4x + 3 * ln(√(1 + x^2) + 1) - 3 * ln(x + √(1 + x^2)) + C2
Where ln represents the natural logarithm and C2 is the constant of integration.
We are given that F(1) = 5. Substituting this into the expression, we get:
5 = 4(1) + 3 * ln(√(1 + 1^2) + 1) - 3 * ln(1 + √(1 + 1^2)) + C2
5 = 4 + 3 * ln(√2 + 1) - 3 * ln(1 + √2) + C2
Now, we can solve this equation to find the value of C2.
d/dx arctan(x) = 1/(1+x^2)
now do the integral and plug in (1,5) to find C