Integral of Sin2x

In mathematics, integration is the process of finding a function g(x) whose derivative, Dg(x), is equal to a given function f(x). This is represented by the integral symbol “∫” as in ∫f(x), which is commonly referred to as the function’s indefinite integral.

Sin 2x Integral:

∫ sin 2x dx is the integral of sin 2x, and its value is -(cos 2x) / 2 + C, where ‘C’ is the integration constant.

Proof :

For proving this, we use the integration by substitution method. We’ll use the assumption that 2x = u for this. Then 2 dx = du (or) dx = du/2 comes into play. By substituting these values for integral ∫ sin 2x dx:

∫ sin 2x dx = ∫ sin u (du/2)

= (1/2) ∫ sin u du

Integral of sin x is -cos x + C. So,using it we have :

= (1/2) (-cos u) + C

Substituting u = 2x we have,

∫ sin 2x dx = -(cos 2x) / 2 + C

Definite Integral of Sin 2x

An indeterminate integral with lower and upper boundaries is called a definite integral. To assess a definite integral, we replace the upper bound and lower bound in the value of the indefinite integral and then remove them in the same sequence, according to the fundamental theorem of calculus. We can omit the integration constant when assessing a definite integral. We will now calculate some definite integrals of integral sin 2x dx.

From 0 to pi/2, the integral of Sin 2x

∫0π/2 sin 2x dx = (-1/2) cos (2x) |π/20

= (-1/2) [cos 2(π/2) – cos 2(0)]

= (-1/2) (-1 – 1)

= (-1/2) (-2)

= 1

Hence, the integral of sin 2x from 0 to pi/2 is 1.

Integral of Sin 2x From 0 to pi

∫0π sin 2x dx = (-1/2) cos (2x) |π0

= (-1/2) [cos 2(π) – cos 2(0)]

= (-1/2) (1 – 1)

= 0

Hence.the integral of sin 2x from 0 to pi is 0.

Examples :

 1. Find the  Integration of Sin(2x+1).

We can write Integration of sin(2x+1) as: ∫ sin(2x + 1)dx

It is known that, ∫ sin2x dx = -(½) cos2x + C

So, ∫ sin(2x + 1) dx = -(½) cos(2x+1) + C

 2.Find the Integration of Sin2x/1+cosx.

∫Sin2x/1+cosx

= ∫ (sin2x)/(1 + cos x) dx

By using sin2x formula, i.e. sin2x = 2 sinx cosx

= ∫ (2 sinx cosx)/(1 + cosx) dx

= 2 ∫[cosx/(1 + cos x)] sinx dx

Let u = cos x

du = -sinx dx

Susbtituting these values, we get;

= 2 ∫[u/(1 + u)] (-du)

= -2 ∫ (u + 1 – 1)/(u + 1) du

= -2 ∫ (u + 1)/(u + 1) du + 2 ∫ 1/(u + 1) du

= -2 ∫ 1 du + 2 log|u + 1|

= -2u + 2log|u + 1| + C

= -2 cosx + 2log|cosx + 1| + C

Formulas for the integration of Trigonometric Functions :

1.∫sin x dx = -cos x + C

2.∫cos x dx = sin x + C

3.∫tan x dx = ln|sec x| + C

4.∫sec x dx = ln|tan x + sec x| + C

5.∫cosec x dx = ln|cosec x – cot x| + C = ln|tan(x/2)| + C

6.∫cot x dx = ln|sin x| + C

7.∫sec2x dx = tan x + C

8.∫cosec2x dx = -cot x + C

9.∫sec x tan x dx = sec x + C

10.∫cosec x cot x dx = -cosec x + C

11.∫sin kx dx = -(cos kx/k) + C

12.∫cos kx dx = (sin kx/k) + C

Examples :

1. Find the Integration of Sin x Cos x by Cos x substitution.

The following formulas will be used  to find the integral of sin x cos x:

d(cos x)/dx = -sin x

∫xn dx = xn+1/(n + 1) + C

Let cos x = v, then we have -sin x dx = dv ⇒ sin x dx = -dv. By Using the above formulas, we have

∫ sin x cos x dx = ∫-vdv

= -v2/2 + C

⇒ ∫ sin x cos x dx = (-1/2) cos2x + C

Hence by substituting cos x, we obtained the integration of sin x cos x.

2. Find the Integral of Sin2x  by Using Double Angle Formula of Cos.

We utilise the double angle formula of cos to obtain the integral of Sin2x   . cos 2x = 1 – 2 sin2x is one of the formula for cos2x. By solving this for sin2x, we get sin2x = (1 – cos 2x) / 2. This is what we use to determine∫ sin2x dx. Then we get

∫ sin2x dx = ∫ (1 – cos 2x) / 2 dx

= (1/2) ∫ (1 – cos 2x) dx

= (1/2) ∫ 1 dx – (1/2) ∫ cos 2x dx

We know that ∫ cos 2x dx = (sin 2x)/2 + C. So

∫ sin2x dx = (1/2) x – (1/2) (sin 2x)/2 + C (or)

∫ sin2x dx = x/2 – (sin 2x)/4 + C

Conclusion :

Integration is mostly used to compute the volumes of three-dimensional objects and to calculate the areas of two-dimensional regions. Finding the area of the curve with respect to the x-axis is the same as finding the integral of a function with respect to the x-axis.