Integrate 4,^~^1(5×^2-8×+5)d×
To integrate the expression 4,^~^1(5×^2 - 8× + 5)d×, you can begin by expanding the power of the polynomial:
4,^~^1(5x^2 - 8x + 5) = 4,^~^1(5x^2) - 4,^~^1(8x) + 4,^~^1(5)
Now, we can integrate each term separately:
∫ 4,^~^1(5x^2) d× = 4,^~^1 * (5/3)x^3 + C1, where C1 is the constant of integration.
∫ 4,^~^1(-8x) d× = 4,^~^1 * (-4)x^2 + C2, where C2 is another constant of integration.
∫ 4,^~^1(5) d× = 4,^~^1 * 5x + C3, where C3 is another constant of integration.
Finally, the integral of the original expression is:
4,^~^1 * (5/3)x^3 - 4,^~^1 * 4x^2 + 4,^~^1 * 5x + C,
where C is the constant of integration for the entire expression.
To integrate the expression 4x^(-1)(5x^2 - 8x + 5)dx, follow these steps:
Step 1: Distribute the 4x^(-1) through the expression inside the parentheses:
4x^(-1)(5x^2 - 8x + 5) = 20x + (-32x^0) + 20x^(-1)
Simplified expression: 20x - 32 + 20/x
Step 2: Integrate each term separately:
∫(20x - 32 + 20/x) dx
= ∫20x dx - ∫32 dx + ∫20/x dx
Step 3: Integrate each term using the power rule of integration:
∫20x dx = 10x^2 + C1
∫32 dx = 32x + C2
∫20/x dx = 20ln|x| + C3
Final result: 10x^2 + 32x + 20ln|x| + C, where C = C1 + C2 + C3, is the indefinite integral of 4x^(-1)(5x^2 - 8x + 5)dx.
To integrate the expression 4^(5×^2-8×+5) with respect to d, we need to use advanced techniques in calculus called substitution or integration by parts.
Let's break it down step by step:
Step 1: Rewrite the expression
The first step is to rewrite the expression using the power rule of exponents. The expression 4^(5×^2-8×+5) can be rewritten as (4^5×^2) × (4^-8×) × (4^5).
Step 2: Integration by parts
Now we can use the technique of integration by parts. The formula for integration by parts states that ∫u * dv = u * v - ∫v * du, where u and v are functions and du and dv represent their differentials.
Let's assign:
u = 4^5×^2
dv = (4^-8×) * (4^5) * d×
Now we need to find the differentials du and v for these assignments:
du = d(4^5×^2) = 2×5×4^5×^(2-1) d× = 10×4^5×^d×
v = ∫dv = ∫(4^-8×) * (4^5) * d× = ∫4^(-8×+5) d×
Step 3: Simplify and integrate
Now, we integrate the expression:
∫4^(-8×+5) d×
To integrate this expression, we can use the power rule for integration:
The integral of x^n with respect to x is (1/(n+1)) * x^(n+1) + C, where C is the constant of integration.
Using this rule, the integral of 4^(-8×+5) is (1/(-8×+6)) * 4^(-8×+6) + C.
Step 4: Combine the results
Now, we have all the pieces to put everything together:
∫4^(5×^2-8×+5) d× = (4^5×^2) * [(1/(-8×+6)) * 4^(-8×+6) + C]
Simplifying further, we have:
∫4^(5×^2-8×+5) d× = (1/(-8×+6)) * 4^(5×^2-8×+6) + C
And that's the final result.