Find the indefinite integral and check the result by differentiation.
∫3√((1-2x²)(-4x))dx
To find the indefinite integral of the given expression, follow these steps:
Step 1: Simplify the expression
First, you need to simplify the expression under the square root. Expand the product inside the square root.
(1 - 2x²)(-4x) = -4x + 8x³
So, the simplified expression inside the square root is -4x + 8x³.
Step 2: Rewrite the expression
Rewrite the expression √(-4x + 8x³) as (-4x + 8x³)^(1/2).
Step 3: Find the indefinite integral
Now that you have the expression in the correct form, you can find the indefinite integral. Integrate term by term.
∫3√((1-2x²)(-4x))dx = 3∫(-4x + 8x³)^(1/2) dx
Step 4: Apply the power rule for integration
Apply the power rule for integration, which is ∫x^n dx = (x^(n+1))/(n+1), for non-negative integer values of n.
∫(-4x + 8x³)^(1/2) dx = (2/3)(-4x + 8x³)^(3/2) + C
Therefore, the indefinite integral of 3√((1-2x²)(-4x))dx is (2/3)(-4x + 8x³)^(3/2) + C, where C is the constant of integration.
To check the result by differentiation, differentiate the obtained indefinite integral and see if you get back the original expression.
Let's differentiate (2/3)(-4x + 8x³)^(3/2) + C with respect to x.
(d/dx)[(2/3)(-4x + 8x³)^(3/2) + C]
Applying the chain rule, the derivative of (-4x + 8x³)^(3/2) is (3/2)(-4x + 8x³)^(1/2)(-4 + 24x²).
(d/dx)[(2/3)(-4x + 8x³)^(3/2) + C] = (2/3)(3/2)(-4x + 8x³)^(1/2)(-4 + 24x²)
Simplifying further, we get
= (2)(-4x + 8x³)^(1/2)(-4 + 24x²)
= -8(-4x + 8x³)^(1/2)(-4 + 24x²)
= 8(4x - 8x³)^(1/2)(4 - 24x²)
The result obtained after differentiation does not match the original expression 3√((1-2x²)(-4x)). As a result, it indicates that there might have been an error in the calculation.