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Integration of 2x
Integration of 2x is the process of finding the integral of 2x with respect to x. Using the integration rules such as integral of scalar multiple of a function and power rule of integration, we can determine the integration of 2x. As we know, integration is the reverse process of differentiation and hence, we can say that the integral of 2x is the antiderivative of 2x. Mathematically, the integration of 2x is written as ∫2x dx = x^{2} + C, where C is the integration constant.
As we proceed, in this article, we will discover the integral of 2x and its formula. We will also determine the definite integration of 2x using its formula along with some examples for a better understanding of the concept.
1.  What is the Integration of 2x? 
2.  Integration of 2x Formula 
3.  Integral of 2x Proof 
4.  Integration of 2x dx From 10 to 13 
5.  FAQs on Integration of 2x 
What is the Integration of 2x?
The integration of 2x in calculus is equal to x square plus the constant of integration which is symbolically written as ∫2x dx = x^{2} + C, where ∫ is the symbol of the integral, dx shows that the integration of 2x is with respect to the variable x and C is the constant of integration. We can find the formula for the integral of 2x using various integration rules. Let us go through the formula of the integration of 2x in the next section.
Integration of 2x Formula
The formula for the integration of 2x is given by, ∫2x dx = x^{2} + C, with C as the integration constant. We can evaluate the integral of 2x using the antiderivative rules, namely the antiderivative rule for scalar multiplication of function and the antiderivative power rule. The image given below shows the formula for the integration of 2x:
Integral of 2x Proof
In this section, we will derive the integration of 2x using the antiderivative rules. We will use the following integration rules to compute the integral of 2x:
 ∫x^{n} dx = x^{n+1}/(n + 1), where n ≠ 1 → This rule is called the Power Rule of Integration
 ∫kf(x) dx = k ∫f(x) dx, where is k is a real number. → This rule is called the Antiderivative Rule for Scalar Multiple of Function
Using the above rules of integration, we have
∫2x dx = 2 ∫x dx
= 2 ∫x^{1} dx
= 2 [x^{1+1}/(1 + 1)] + C
= 2 x^{2}/2 + C
= x^{2} + C (Cancelling out the 2s in the numerator and denominator)
Hence, we have proved that the integration of 2x is equal to x^{2} + C, where C is the integration constant.
Integration of 2x dx From 10 to 13
Definite integration is the value that is obtained by taking the difference of the antiderivative of the function at the upper limit and lower limit. Now that we know the formula for the integration of 2x which is given by x^{2} + C, next, we will evaluate the definite integral of 2x with limits ranging from 10 to 13. Therefore, we have
\(\begin{align}\int_{10}^{13}2x \ dx &=\left [ x^2+C \right ]_{10}^{13}\\&=(13^2+C)(10^2+C)\\&=169100\\&=69 \end{align}\)
Hence, the integral of 2x with limits ranging from 10 to 13 is equal to 69.
Important Notes on Integration of 2x
 The integration of 2x is written as ∫2x dx = x^{2} + C, where C is the integration constant.
 Using the integration rules  Integral of scalar multiple of a function and power rule of integration, we can determine the integration of 2x.
 The integration of 2x with limits from a to b is given by [(b^{2} + C)(a^{2} + C)]. Hence, the integral of 2x with limits ranging from 10 to 13 is equal to 69.
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Integration of 2x Examples

Example 1: Evaluate the integration of 2x dx limit 1 to 12.
Solution: To determine the value of the integral of 2x with limits from 1 to 12, we will use the formula for the integration of 2x which is ∫2x dx = x^{2} + C.
\(\begin{align}\int_{1}^{12}2x \ dx &=\left [ x^2+C \right ]_{1}^{12}\\&=(12^2+C)(1^2+C)\\&=1441\\&=143 \end{align}\)
Answer: Hence, the integral of 2x with limits from 1 to 12 is equal to 143.

Example 2: Compute the value of integration of 2x by (1 + x^{2}).
Solution: To find the integral of 2x / (1 + x^{2}), we will use the substitution method.
Assume 1 + x^{2} = u ⇒ 2x dx = du, therefore we have
∫[2x / (1 + x^{2})] dx = ∫du/u
= ln u + C
= ln 1 + x^{2} + C
Answer: Hence, the integration of 2x by (1 + x^{2}) is ln 1 + x^{2} + C.
FAQs on Integration of 2x
What is Integration of 2x in Calculus?
Integration of 2x is the process of finding the integral of 2x with respect to x. Mathematically, the integration of 2x is written as ∫2x dx = x^{2} + C, where C is the integration constant.
What is the Formula of Integration of 2x?
The formula for the integration of 2x is given by, ∫2x dx = x^{2} + C, with C as the integration constant.
How Do You Solve the Integral of 2x?
Using the integration rules such as integral of scalar multiple of a function and power rule of integration, we can determine the integration of 2x.
What is the Definite Integration of 2x with Limits from 10 to 13?
The integral of 2x with limits ranging from 10 to 13 is equal to 69. This can be calculated using the formula ∫2x dx = x^{2} + C.
How to Find the Integration of 2x + 3?
The integration of 2x + 3 can be determined as, ∫(2x + 3)dx = 2 ∫x dx + 3 ∫dx = x^{2} + 3x + C, where C is the integration constant. The integration rules used to find this integral of 2x + 3 are product rule, integral of constant and integration rule of scalar multiple of function.
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