Evaluating Integrals Using Substitution Techniques
In this article, we will walk through the process of evaluating an integral using substitution techniques. Specifically, we will examine the integral:
I ∫24 (x2 - x) / √(2x 1) dx
Step 1: Substitution Method
To simplify this integral, we will use a substitution. Let:
u 2x 1
Then, we differentiate u with respect to x:
du/dx 2 implies dx du / 2
Step 2: Changing the Limits of Integration
Next, we need to change the limits of integration:
When x 2, u 2(2) 1 5 When x 4, u 2(4) 1 9Step 3: Expressing x in Terms of u
We express x in terms of u:
x (u - 1) / 2
Step 4: Substituting into the Integral
Now, we substitute x into the given expression:
(x2 - x) ((u - 1)2 / 4) - ((u - 1) / 2)
Simplifying, we get:
(u2 - 1) / 4
Substituting everything into the integral, we have:
I ∫59 ((u2 - 1) / 4) / √u * (du / 2)
Which simplifies to:
I 1/8 ∫59 (u2 - 1) / √u du
Step 5: Splitting the Integral
Now, we split the integral:
I 1/8 ( ∫59 u3/2 du - ∫59 u-1/2 du)
Step 6: Calculating Each Integral Separately
For ∫ u3/2 du: Integral: u5/2 / (5/2) 2/5 u5/2 Evaluating from 5 to 9: 2/5 (95/2 - 55/2) Calculating: 2/5 (243 - 25√5) For ∫ u-1/2 du: Integral: 2u1/2 Evaluating from 5 to 9: 2 (3 - √5)Step 7: Substituting Back into the Expression for I
Combining the terms gives:
I 1/8 ( 2/5 (243 - 25√5) - 2 (3 - √5) )
Further simplifying and finding a common denominator 5:
I 1/8 ( (486 - 50√5 - 30 10√5) / 5 ) 1/8 ( (456 - 40√5) / 5 ) (456 - 40√5) / 40 (114 - 10√5) / 10
Final Result
The final result for the integral is:
I (114 - 10√5) / 40
Alternative Substitution Approach
Alternatively, we can use the substitution: 2x 1 t2.
This implies:
dx t dt and 2x 1 t2 implies 4x2 - 4x - 1 t4 implies x2 - x (t4 - 1) / 4
When x 2, t √5 When x 4, t 3Thus, the integral becomes:
I ∫√53 (t4-1 / 4) / t * t dt 1/4 ∫√53 (t4 - 1) dt
After evaluating, we have:
I 1/4 ( (t5/5) - t )√53 1/4 ( (35/5) - 3 - ( (5√5)/5 ) √5 1/4 ( 228/5 - 4√5) 57/5 - √5
This approach also leads to the same result:
boxed{I 57/5 - √5}