Questions: ∫(7x^4 - 6x^2 + 6) dx

Transcript text: \[ \int\left(7 x^{4}-6 x^{2}+6\right) d x \]

Solution

Solution Steps

Step 1: Identify the coefficients and the degree of the polynomial

The coefficients are [7, 0, -6, 0, 6] and the degree is 4.

Step 2: Apply the power rule of integration to each term

The power rule of integration states that for any term $ax^n$, its integral is given by: $$\int(ax^n)dx = \frac{a}{n+1}x^{n+1} + C$$ Applying this rule to each term of the polynomial and summing the results gives the general solution to the indefinite integral of the polynomial.

Step 3: Calculate the integral for each term

For the term with coefficient 7 and power 4, the integral is 1.4x^5. For the term with coefficient 0 and power 3, the integral is 0.0x^4. For the term with coefficient -6 and power 2, the integral is -2.0x^3. For the term with coefficient 0 and power 1, the integral is 0.0x^2. For the constant term 6, the integral is 6x.

Final Answer:

The indefinite integral of the polynomial is 1.4x^5 + 0.0x^4 - 2.0x^3 + 0.0x^2 + 6x + C.

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