3. [tex]\int(3x - 2)^6 dx[/tex]
Ans: [tex](1/504)(3x - 2)^7 + C[/tex]
4. [tex]\int(x(x-4))/x dx[/tex]
Ans: [tex](x^2 + 4x + 8)/x + C[/tex]
Answer (2)
Let's address each of these integration problems step by step.
Integral of ( 3 x − 2 ) 6 dx :
To solve this, we can use the substitution method.
Let u = 3 x − 2 . Then, the derivative d u = 3 d x , or d x = 3 1 d u .
Substitute into the integral:
∫ ( 3 x − 2 ) 6 d x = ∫ u 6 3 1 d u = 3 1 ∫ u 6 d u
Now integrate u 6 :
= 3 1 ⋅ ( 7 u 7 ) + C = 21 1 u 7 + C
Substitute back u = 3 x − 2 :
= 21 1 ( 3 x − 2 ) 7 + C
It seems there is a mismatch with the given answer. Upon re-evaluation, this interpretation leads to:
= 21 1 ( 3 x − 2 ) 7 + C
Integral of x x ( x − 4 ) dx :
First, simplify the expression:
x x ( x − 4 ) = x − 4
Now, integrate each term separately:
∫ ( x − 4 ) d x = ∫ x d x − ∫ 4 d x
The integrals become:
= 2 x 2 − 4 x + C
Ensure proper understanding:
-Distribution and simplification clarify the expression simplifies to x − 4 , leading to a straightforward integration.
Notice that if simplified to a single expression x x 2 − 4 x = x − 4 , the integral yields:
2 x 2 − 4 x + C
Here, the integration performs detailed confirmation with basic simplification steps ensuring clarity.
The first integral ∫ ( 3 x − 2 ) 6 d x is solved through substitution to yield 21 1 ( 3 x − 2 ) 7 + C . The second integral simplifies first to x − 4 , which integrates to 2 x 2 − 4 x + C . Both integrals follow standard integration techniques that include substitution and simplification.
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