Integration of x^2
The integration of x^2 is equal to x3/3 + C, where C is the constant of integration. As we proceed with the evaluation of the integral of x^2, let us recall the meaning of integration. Integration is the reverse process of differentiation and that is why it is also called the process of antidifferentiation. To determine the integration of x^2 (that is, integral of x2), we need to find an arbitrary function whose derivative is x2. We can calculate this integral using the power rule of integration. The formula for the integral of x2 is written as ∫x2 dx = x3/3 + C.
Let us calculate the integration of x2 using different methods of integration including the integration by parts method and power rule method of integration. We will also solve examples and determine integrals of functions involving x2 for a better understanding of the concept.
| 1. | What is Integration of x^2? |
| 2. | Integral of x2 Formula |
| 3. | Integration of x^2 Proof |
| 4. | Integration of x^2 Using Integration by Parts |
| 5. | Definite Integration of x^2 |
| 6. | FAQs on Integration of x^2 |
What is Integration of x^2?
The integration of x2 is equal to x3/3 + C. The integral of a function gives the area under the curve of the function. Therefore, the integral of x^2 gives the area under the curve of the function f(x) = x2. Mathematically, we can write the integration of x^2 as, ∫x2 dx = x3/3 + C, where
- C is the constant of integration.
- ∫ is the symbol of an integral
- dx shows that the integration is with respect to the variable x.
To determine the integration of x^2, we need to find the function whose derivative is equal to x2. So, we need to find the question mark in the equation d(?)/dx = x2. Using the power rule of differentiation, we know that d(xn)/dx = nxn-1. Using this formula, we know that the derivative of x3 is equal to 3x2. To get the derivative equal to x2, we divide x3 by 3. So, the derivative of x3/3 is equal to x2. We add the integration constant to all indefinite integrals in calculus. Therefore, the question mark in the equation d(?)/dx = x2 is equal to x3/3 + C, where C is the integration constant.
Integral of x2 Formula
The formula for the integration of x^2 is given by, ∫x2 dx = x3/3 + C. We can evaluate this integral using the power rule of integration. To verify the result, we can also calculate the integral of x^2 using the integration by parts method. The image given below shows the formula for the integration of x2:
Integration of x^2 Proof
Now that we know that the integral of x^2 is equal to x3/3 + C, we will prove it using the power rule of integration. According to the power rule of integration, we have the formula ∫xn dx = xn+1/(n+1) + C, where C is the integration constant. Now, substituting n = 3 into this formula, we have
∫x2 dx = x2+1/(2 + 1) + C
= x3/3 + C
Hence, we have proved that the integral of x^2 is equal to x3/3 + C.
Integration of x^2 Using Integration by Parts
To verify the integral of x^2 derived in the previous section, we can calculate the integral using the integration by parts method. We will use the formula ∫f(x) g(x) dx = f(x) ∫g(x) dx - ∫[f'(x) ∫g(x) dx] dx. Here, substitute f(x) = x2 and g(x) = 1. Also, we can write x2 as x2 = 1.x2. We will use the following formulas to verify the integration of x^2.
- Integral of 1: ∫1dx = x + C
- Derivative of x2: d(x2)/dx = 2x
So, we have
∫x2 dx = ∫1.x2 dx
⇒ ∫x2 dx = x2 ∫1dx - ∫[d(x2)/dx × ∫1dx] dx
⇒ ∫x2 dx = x2(x) - ∫[2x × x] dx + K
⇒ ∫x2 dx = x3 - 2 ∫x2 dx + K
⇒ ∫x2 dx + 2 ∫x2 dx = x3+ K
⇒ 3 ∫x2 dx = x3 + K
⇒ ∫x2 dx = x3/3 + C, where C = K/3
Hence, we have verified that the integration of x^2 is equal to x3/3 + C.
Definite Integration of x^2
The definite integral of a function is a real number that is given by substituting the limits (upper limit and lower limit) of the integration into the formula of the integral. Suppose, we have a definite integral of x^2 with a lower limit a and an upper limit b. Then, it is written as, a ∫b x2dx. We can find the value of this definite integral by substituting the limits a and b into the formula of the integral of x^2 and subtracting them. We have
a ∫b x2dx = [x3/3 + C]ab
= (b3/3 + C) - (a3/3 + C)
= b3/3 + C - x3/3 - C
= b3/3 - a3/3
Hence, the definite integral of x^2 with a lower limit a and upper limit b is given by, b3/3 - a3/3, where a, b are real numbers.
Important Notes on Integration of x^2
- The integration of x2 is equal to x3/3 + C, where C is the integration constant.
- We can evaluate the integral of x^2 using the power rule of integration.
- The definite integral of x^2 with a lower limit a and upper limit b is b3/3 - a3/3.
☛ Related Topics:
FAQs on Integration of x^2
What is Integration of x^2 in Calculus?
The integration of x^2 is equal to x3/3 + C, where C is the constant of integration. The integral of x^2 gives the area under the curve of the function f(x) = x2.
What is the Formula for the Integral of x^2?
The formula for the integration of x^2 is given by, ∫x2 dx = x3/3 + C. We can evaluate this integral using the power rule of integration.
What is the Integral of x^2 + lnx?
The integral of x2 + lnx is equal to ∫(x2 + lnx) dx = ∫x2 dx + ∫lnx dx = x3/3 + x ln(x) − x + C. (Because of integral of ln x is x ln(x) − x + C)
How Do You Find the Integration of x^2?
We can evaluate the integration of x2 using the power rule of integration. We can also verify this formula using the integration by parts method of integration. We have ∫x2 dx = x2+1/(2 + 1) + C = x3/3 + C.
What is the Integration of 1/x^2?
The integration of 1/x^2 is given by, ∫(1/x2) dx = ∫x-2 dx = x-2+1/(-2 + 1) + C = -1/x + C. Therefore, the integration of 1/x^2 is equal to -1/x + C.
Is the Integration of x^2 the Same as the Antiderivative of x^2?
Integration is the reverse process of differentiation and that is why it is also called the process of antidifferentiation. Therefore, the integration of x^2 is the same as the antiderivative of x^2.