Evaluate the definite integral.
x sqrt(13 x^2 + 36)dx between (0,1)
x √(13x^2 +36)dx
u = 13x^2 + 36
between (36, 49)
du= 26x
du/26 = x
1/26 (u)^1/2 du
1/26 *2/3 (u)^3/2 to 36 to 49
1/39[ 49^3/2 -36^3/2]
1/39( 343-216) = 127/39
To evaluate the definite integral ∫(0 to 1) x√(13x^2 + 36) dx, we can use the method of integration by substitution. Here's how you can solve it step by step:
Step 1: Choose a substitution
Let's substitute u = 13x^2 + 36. Taking the derivative on both sides, we have du/dx = 26x. Next, we can solve for dx to express it in terms of du: dx = du / (26x).
Step 2: Change the limits
When x = 0, the corresponding value of u can be found by substituting: u = 13(0)^2 + 36 = 36.
When x = 1, the corresponding value of u can be found similarly: u = 13(1)^2 + 36 = 49.
So, the original definite integral can be transformed into the new integral using the substitutions of u and the new limits:
∫(0 to 1) x√(13x^2 + 36) dx = ∫(36 to 49) (√u / 26) du.
Step 3: Integrate the transformed integral
Simplifying the transformed integral, we get:
(1/26) ∫(36 to 49) √u du.
Step 4: Evaluate the integral
To evaluate the integral, we can use the power rule for integration. The integral of u^k is (1/(k+1)) * u^(k+1).
Applying this rule, we integrate √u as follows:
(1/26) * (√u * 2/3) evaluated from 36 to 49.
Now, evaluating at the upper and lower limits:
[(1/26) * (√49 * 2/3)] - [(1/26) * (√36 * 2/3)]
= [(1/26) * (7 * 2/3)] - [(1/26) * (6 * 2/3)]
= [(14/78) - (12/78)]
= 2/78
= 1/39.
Therefore, the value of the definite integral ∫(0 to 1) x√(13x^2 + 36) dx is 1/39.