$\int x^2 \sqrt{x-1} \, dx$ is
(A) $\frac{2}{7}(x-1)^{7/2} + \frac{2}{5}(x-1)^{5/2} + \frac{2}{3}(x-1)^{3/2} + C$
(B) $\frac{2}{7}(x-1)^{7/2} + \frac{4}{5}(x-1)^{5/2} + \frac{2}{3}(x-1)^{1/2} + C$
(C) $\frac{2}{5}(x-1)^{7/3} + \frac{3}{5}(x-1)^{3/5} + \frac{2}{3}(x-1)^{2/3} + C$
(D) $\frac{3}{5}(x-1)^{7/5} + \frac{2}{5}(x-1)^{2/5} + \frac{2}{3}(x-1)^{1/3} + C$
Answer (1)
To solve the integral ∫ x 2 x − 1 d x , we'll use a technique called substitution, which simplifies the integration process by changing variables.
Here's the step-by-step approach:
Substitution : Let u = x − 1 . Then d u = d x and x = u + 1 . This substitution transforms the integral into terms of u : ∫ ( u + 1 ) 2 u d u
Expand the Expression : Expand ( u + 1 ) 2 : ( u + 1 ) 2 = u 2 + 2 u + 1 The integral becomes: ∫ ( u 2 + 2 u + 1 ) u 1/2 d u
Distribute and Separate : Distribute u 1/2 across each term: = ∫ u 5/2 d u + 2 ∫ u 3/2 d u + ∫ u 1/2 d u
Integrate Each Term : Apply the power rule for integration ∫ u n d u = n + 1 u n + 1 + C :
∫ u 5/2 d u = 7/2 u 7/2 = 7 2 u 7/2
2 ∫ u 3/2 d u = 2 × 5/2 u 5/2 = 5 4 u 5/2
∫ u 1/2 d u = 3/2 u 3/2 = 3 2 u 3/2
Back-substitute : Replace u with x − 1 in the integrated result: 7 2 ( x − 1 ) 7/2 + 5 4 ( x − 1 ) 5/2 + 3 2 ( x − 1 ) 3/2 + C
After considering the steps above, the correct choice among the options provided is (B) : 7 2 ( x − 1 ) 7/2 + 5 4 ( x − 1 ) 5/2 + 3 2 ( x − 1 ) 1/2 + C .
