Integral of 2sinx
The integral of 2sinx is equal to 2cosx + C. This integral can be evaluated using the formula for the integral of sin x. We know that integration is the reverse of the process of differentiation and hence, the integral of 2sinx is also called the antiderivative of 2sinx. We know that the integral of sinx is equal to cosx + C. Therefore, the integration of 2sinx is given by 2cosx + C, where C is the constant of integration.
In this article, we will compute the integral of 2sinx and derive its formula. We will also evaluate the definite integration of 2sinx with different limits and go through a few solved examples based on a similar concept for a better understanding.
1.  What is the Integral of 2sinx? 
2.  Integral of 2sinx Formula 
3.  Definite Integration of 2sinx 
4.  FAQs on Integral of 2sinx 
What is the Integral of 2sinx?
The integral of 2sinx can be calculated using the formula for integral of sinx which is given by, ∫sin x dx = cos x + C, where C is the integration constant. Now, to find the integration of 2sinx, we use this formula and mathematically, write the integral of 2sinx as ∫2sinx dx = 2cosx + C, with C as the constant of integration. Let us now explore the formula for the same in the next section.
Integral of 2sinx Formula
As we know that ∫sin x dx = cos x + C, therefore the formula for the integral of 2sinx is written as ∫2sinx dx = 2cosx + C, where ∫ symbol denotes the integration of 2sinx, dx denotes that the variable is x and C is the integration constant. The image below gives the integral of the 2sinx formula:
Integral of 2sinx Proof
Now that we know the integral of 2sinx is equal to 2cosx + C, next in this section, we will prove this result using different formulas of integration. We will use the following antiderivative rules and formulas:
 ∫sinx dx = cosx + C
 ∫kf(x) dx = k ∫f(x) dx
Using the above formulas, we have
∫2sinx dx = 2 ∫sinx dx
= 2[cosx + K]
= 2cosx + 2K
= 2cosx + C, where C = 2K is the integration constant.
Hence, we have proved that the integral of 2sinx is equal to 2cosx + C.
To verify that the integration of 2sinx is 2cosx + C, we can take the derivate of 2cosx + C. We know that the derivative of cos x is equal to sin x, that is, d(cos x)/dx = sin x and the derivative of constant function is equal to zero. Therefore, we have d(2cosx + C)/dx = d(2cosx)/dx + dC/dx = 2d(cosx)/dx + 0 = 2(sinx) = 2sinx. Hence, verified.
Definite Integration of 2sinx
We know that the formula for the integration of 2sinx is equal to 2cosx + C. Now, we will determine the definite integral of 2sinx with limits from 0 to pi. To find this definite integral, we will substitute the limits into the formula of the integral of 2sinx and take their difference.
\(\begin{align}\int_{0}^{\pi}2\sin x \ dx &= \left [ 2 \cos x + C\right ]_{0}^{\pi}\\&=(2\cos \pi+C)(2\cos 0 + C)\\&=2\times (1)+C+2\times1C\\&=2+2\\&=4 \end{align}\)
Hence, the value of the definite integral of 2sinx with limits from 0 to π is equal to 4.
Important Notes on Integral of 2sinx
 The integral of 2sinx is mathematically written as ∫2sinx dx = 2cosx + C.
 We can find the integration of 2sinx using the integral of sinx.
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Integral of 2sinx Examples

Example 1: Evaluate the integral of 2sinx cosx.
Solution: To determine the integral of 2sinx cosx, we will use the sin 2x formula. We know that sin2x = 2sinx cosx and integral of sin 2x is equal to (1/2)cos 2x + C. Hence, we have
∫2sinx cosx dx = ∫sin2x dx
= (1/2) cos2x + C
Answer: The integral of 2sinx cosx is (1/2) cos2x + C.

Example 2: Determine the integral of 2cos^{2}x.
Solution: To find the integral of 2cos^{2}x, we will use the cos2x formula which is cos 2x = 2cos^{2}x  1 ⇒ 2cos^{2}x = cos2x + 1. Therefore, we have
∫2cos^{2}x dx = ∫(cos2x + 1) dx
= ∫cos 2x dx + ∫dx
= (1/2) sin 2x + x + C
Answer: The integral of 2cos^{2}x is (1/2) sin 2x + x + C.
FAQs on Integral of 2sinx
What is the Integral of 2sinx?
The integral of 2sinx is equal to 2cosx + C, where C is the integration constant. This integral can be evaluated using the formula for the integral of sin x.
What is the Formula for the Integration of 2sinx?
The integral of 2sinx can be calculated using the formula for integral of sinx which is given by, ∫2sin x dx = 2cos x + C, where C is the integration constant.
How To Find Integral of 2sinx?
We can find the integral of 2sinx by using the following antiderivative rules and formulas:
 ∫sinx dx = cosx + C
 ∫kf(x) dx = k ∫f(x) dx
Using the above formulas, we have the integral of 2sinx to be equal to 2cosx + C, with C as the integration constant.
What is the Indefinite Integral of 2sinx With Respect to cosx?
The indefinite integral of 2sinx with respect to cosx is determined as ∫2sinx d(cosx) = ∫2sinx (sinx) dx = ∫2sin^{2}x dx =  ∫(1  cos2x) dx = (x  sin2x/2) + C = x + (1/2)sin2x + C. Hence, the integral of 2sinx with respect to cosx is given by x + (1/2)sin2x + C, where C is the constant of integration.
What is the Value of Integral of 2sinx From 0 to pi?
The value of the definite integral of 2sinx with limits from 0 to π is equal to 4. This integral value is determined by substituting the limits into the formula of the integral of 2sinx and subtracting the two.
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