Answer :
To solve the integral and find the value of \(a\) where \(\int_1^a (3x^2 - 2x - 4) \, dx = 10\), follow these steps:
### Step 1: Compute the Indefinite Integral
First, we need to find the indefinite integral of the function \(3x^2 - 2x - 4\).
[tex]\[ \int (3x^2 - 2x - 4) \, dx \][/tex]
We integrate each term separately:
[tex]\[ \int 3x^2 \, dx - \int 2x \, dx - \int 4 \, dx \][/tex]
The integrals of each term are:
[tex]\[ \int 3x^2 \, dx = x^3 \][/tex]
[tex]\[ \int 2x \, dx = x^2 \][/tex]
[tex]\[ \int 4 \, dx = 4x \][/tex]
Combining these results, we get the indefinite integral:
[tex]\[ \int (3x^2 - 2x - 4) \, dx = x^3 - x^2 - 4x + C \][/tex]
### Step 2: Evaluate the Definite Integral
Next, we evaluate this indefinite integral from \(x = 1\) to \(x = a\):
[tex]\[ \left[ x^3 - x^2 - 4x \right]_1^a \][/tex]
This means we substitute \(a\) and 1 into the integrated function:
[tex]\[ = (a^3 - a^2 - 4a) - (1^3 - 1^2 - 4 \cdot 1) \][/tex]
Simplify the expression:
[tex]\[ = (a^3 - a^2 - 4a) - (1 - 1 - 4) \][/tex]
[tex]\[ = a^3 - a^2 - 4a - (-4) \][/tex]
[tex]\[ = a^3 - a^2 - 4a + 4 \][/tex]
So the definite integral is:
[tex]\[ = a^3 - a^2 - 4a + 4 \][/tex]
### Step 3: Set Up the Equation
We are given that this integral equals 10:
[tex]\[ a^3 - a^2 - 4a + 4 = 10 \][/tex]
### Step 4: Solve for \(a\)
Rearrange the equation to solve for \(a\):
[tex]\[ a^3 - a^2 - 4a + 4 - 10 = 0 \][/tex]
[tex]\[ a^3 - a^2 - 4a - 6 = 0 \][/tex]
To find the roots of the polynomial equation \(a^3 - a^2 - 4a - 6 = 0\), we can use various methods such as factoring, if possible, or applying the cubic formula. However, the roots of this cubic polynomial are:
[tex]\[ a = 3, \quad a = -1 - i, \quad a = -1 + i \][/tex]
Since we are dealing with the definite integral from 1 to \(a\) which resembles the context of a real number, we consider the real solution:
[tex]\[ a = 3 \][/tex]
### Conclusion
Therefore, \(a\) that satisfies the given definite integral equation is:
[tex]\[ \boxed{3} \][/tex]
### Step 1: Compute the Indefinite Integral
First, we need to find the indefinite integral of the function \(3x^2 - 2x - 4\).
[tex]\[ \int (3x^2 - 2x - 4) \, dx \][/tex]
We integrate each term separately:
[tex]\[ \int 3x^2 \, dx - \int 2x \, dx - \int 4 \, dx \][/tex]
The integrals of each term are:
[tex]\[ \int 3x^2 \, dx = x^3 \][/tex]
[tex]\[ \int 2x \, dx = x^2 \][/tex]
[tex]\[ \int 4 \, dx = 4x \][/tex]
Combining these results, we get the indefinite integral:
[tex]\[ \int (3x^2 - 2x - 4) \, dx = x^3 - x^2 - 4x + C \][/tex]
### Step 2: Evaluate the Definite Integral
Next, we evaluate this indefinite integral from \(x = 1\) to \(x = a\):
[tex]\[ \left[ x^3 - x^2 - 4x \right]_1^a \][/tex]
This means we substitute \(a\) and 1 into the integrated function:
[tex]\[ = (a^3 - a^2 - 4a) - (1^3 - 1^2 - 4 \cdot 1) \][/tex]
Simplify the expression:
[tex]\[ = (a^3 - a^2 - 4a) - (1 - 1 - 4) \][/tex]
[tex]\[ = a^3 - a^2 - 4a - (-4) \][/tex]
[tex]\[ = a^3 - a^2 - 4a + 4 \][/tex]
So the definite integral is:
[tex]\[ = a^3 - a^2 - 4a + 4 \][/tex]
### Step 3: Set Up the Equation
We are given that this integral equals 10:
[tex]\[ a^3 - a^2 - 4a + 4 = 10 \][/tex]
### Step 4: Solve for \(a\)
Rearrange the equation to solve for \(a\):
[tex]\[ a^3 - a^2 - 4a + 4 - 10 = 0 \][/tex]
[tex]\[ a^3 - a^2 - 4a - 6 = 0 \][/tex]
To find the roots of the polynomial equation \(a^3 - a^2 - 4a - 6 = 0\), we can use various methods such as factoring, if possible, or applying the cubic formula. However, the roots of this cubic polynomial are:
[tex]\[ a = 3, \quad a = -1 - i, \quad a = -1 + i \][/tex]
Since we are dealing with the definite integral from 1 to \(a\) which resembles the context of a real number, we consider the real solution:
[tex]\[ a = 3 \][/tex]
### Conclusion
Therefore, \(a\) that satisfies the given definite integral equation is:
[tex]\[ \boxed{3} \][/tex]
