the integral 49dx/x^2(x^2+36)^(1/2)
To solve the integral ∫ 49dx / (x^2(x^2 + 36)^(1/2)), we can use substitution. Let's substitute u = x^2 + 36. Then, du = 2x dx.
Now, rewrite the integral using the substitution:
∫ 49dx / (x^2(x^2 + 36)^(1/2)) = ∫ 49dx / (u^(1/2) - 36)
Next, substitute the value of u and du back into the integral:
∫ (49 / 2x) du / (u^(1/2) - 36)
Now, let's simplify the integral:
∫ (49 / 2x) du / (u^(1/2) - 36) = (49 / 2) ∫ du / (u^(1/2) - 36)
At this point, we have a simplified integral that we can solve. The integral involves an expression in the denominator, which can be expressed as a difference of squares. Let's rewrite it as a difference of squares:
u^(1/2) - 36 = √u - √(36^2) = √u - 6 * 6 = (√u - 6)(√u + 6)
Now, we can rewrite the integral using partial fraction decomposition:
(49 / 2) ∫ du / ((√u - 6)(√u + 6)) = (49 / 2) ∫ (A / (√u - 6) + B / (√u + 6)) du
To find the values of A and B, we can multiply both sides of the equation by the denominator (√u - 6)(√u + 6) and equate the numerators:
1 = A(√u + 6) + B(√u - 6)
Solve this equation to find the values of A and B, and substitute them back into the integral.
Finally, you can integrate the two terms separately and find the antiderivative. Evaluate the integral at the limits of your choice to find the definite integral if necessary.