Integral of Sin4x
The formula for the integral of sin4x is given by (1/4) cos 4x + C, where C is the constant of integration. Integration is the process of finding the antiderivative of a function as finding an integral is a reverse process of finding the derivative of a function. Hence, the integration of sin4x is the same as finding the antiderivative of sin 4x. Integration of sin4x can be calculated using different methods.
In this article, we will determine the integral of sin4x using different methods such as substitution method, its definite integral, and formula along with the integral of sin^4x. We will also determine the integral of sin4x combined with other functions and solve a few examples for a better understanding of the concept.
What is Integration of Sin4x?
Sin 4x is a trigonometric function of sine with an angle of 4x. Integration of sin 4x can be calculated using different methods such as the substitution method. The integration of sin 4x is equal to the negative of onefourth of the cosine of the angle 4x plus the constant of integration which is mathematically written as ∫sin 4x dx = (1/4) cos 4x + C, where C is the constant of integration, dx indicates the integration of sin 4x w.r.t. x and ∫ is the symbol of integration. Let us now go through the formula for integration of sin 4x.
Integral of Sin4x Formula
The formula of integration of sin 4x is written as ∫sin 4x dx = (1/4) cos 4x + C, where the mathematical symbols ∫ denote the integration, dx denotes the variable of integration, and C is the constant of integration. The image given below shows the integration of sin4x formula:
Integration of Sin4x Using Substitution Method
Now, that we know that the integration of sin 4x is ∫sin 4x dx = (1/4) cos 4x + C, we will prove this using the substitution method. For trigonometric function sin 4x, we can assume 4x = u, and differentiating both sides, we get
4dx = du
⇒ dx = (1/4) du
∫sin 4x dx = ∫ sin u (1/4) du
= (1/4) ∫ sin u du
= (1/4) (cos u) + C [Because Integral of sin x is equal to cos x + C]
= (1/4) cos 4x + C
Hence we have proved the integration of sin 4x using the substitution method.
Integral of Sin^4x
In this section, we will evaluate the integral of sin^4x (that is, sin^{4}x). To do so, we will use the cos2x formula in terms of sine only. We know that cos 2x = 1  2sin^{2}x which implies sin^{2}x = (1  cos2x) / 2. Also, we can write sin^{4}x = (sin^{2}x)^{2}. Let, I = ∫ sin^{4}x dx. Here, we use the power reduction formula to integrate sin^{4}x dx.
I = ∫ (sin^{2}x)^{2 }dx
Put (sin^{2}x)^{2 }= {(1 − cos2x) / 2}^{2} [ Since, 2sin^{2}x = 1  cos2x]
= ∫ {(1 − cos2x) /2 }^{2 }dx
= (1/4) ∫ (1 − cos2x)^{2 }dx
= (1/4) ∫ (1− 2cos2x + cos^{2}2x) dx [Since, (a  b)^{2} = a^{2}  2ab + b^{2}]  (1)
Again, we use the power reduction formula for cos^{2}(2x)
cos^{2}(2x) = (1 + cos(4x)) / 2 [Since, 2cos^{2}x = cos2x + 1]  (2)
Substituting (2) in (1) we get,
= (1/4) ∫ [1  2cos2x + {(1 + cos(4x)) / 2 }] dx
= (1/4) ∫ dx  (2/4) ∫ cos2x dx + (1/8) ∫ (1 + cos(4x)) dx
= (x/4)  (1/2) (sin(2x)/2) + (x/8) + (1/8) (sin(4x)/4) + C
On evaluating,
= (3/8)x + (1/32) sin(4x) − (1/4) sin(2x) + C
Hence, the integral of sin^4x is equal to (3/8)x + (1/32) sin(4x) − (1/4) sin(2x) + C.
Integration of Sin4x By Sin x
Next, we will prove the integration of sin 4x / sin x using sin 2A formula. The trigonometric formulas that we will use to determine the integral of sin 4x are:
 sin 2A = 2 sin A cos A
 cos A cos B = (1/2)[cos (A + B) + cos (A  B)]
∫(sin 4x / sin x) dx = ∫(2 sin 2x cos 2x) / sin x dx
= 2 ∫(2 sin x cos x cos 2x ) / sin x dx
= 4 ∫cos x cos 2x dx
= 4 ∫ (1/2)[cos (x + 2x) + cos (x  2x)] dx
= 2 ∫(cos 3x + cos x) dx
= 2 [(1/3) sin 3x + sin x] + C
Hence the integration of sin 4x upon sin x is 2 [(1/3) sin 3x + sin x] + C, where C is the constant of integration.
Integration of Sin4x From 0 to Pi
We know that the integration of sin 4x is (1/4) cos 4x + C. Now, we will calculate the definite integral of sin 4x with limits from 0 to π. Definite integration of sin 4x from 0 to pi will give a finite value.
\(\begin{align} \int_{0}^{\pi} \sin 4x \ dx &= \left [ \frac{1}{4}\cos 4x + C\right ]_0^\pi \\&=\left ( \frac{1}{4}\cos 4\pi + C \right )\left ( \frac{1}{4}\cos 0 + C \right )\\&=\frac{1}{4}+C+\frac{1}{4}C\\&=0\end{align}\)
Therefore, the definite integration of sin 4x from 0 to pi is equal to 0.
Important Notes on Integration of Sin 4x
 The formula of integration of sin 4x is written as ∫sin 4x dx = (1/4) cos 4x + C.
 Integration of sin 4x can be calculated using the substitution method.
☛ Related Topics:
Integration of Sin4x Examples

Example 1: Calculate the integration of sin4x cos4x.
Solution: To determine the integration of sin 4x cos 4x, we will use the trigonometric formula sin 2A = 2 sin A cos A.
∫sin 4x cos 4x dx = (1/2) ∫ 2 sin 4x cos 4x dx
= (1/2) ∫ sin 8x dx
= (1/2) (1/8) cos 8x + C
= (1/16) cos 8x + C
Answer: Hence the integration of sin 4x cos 4x is equal to (1/16) cos 8x + C

Example 2: Determine the integration of sin 4x by cos 2x.
Solution: To find the integration of sin 4x by cos 2x, we will use the trigonometric formula sin 2A = 2 sin A cos A.
∫(sin 4x / cos 2x) dx = ∫(2 sin 2x cos 2x / cos 2x) dx
= ∫ 2 sin 2x dx
= 2 (1/2) cos 2x + C
= cos 2x + C
Answer: The integration of sin 4x upon cos 2x is cos 2x + C.

Example 3: Evaluate the definite integral of sin4x from 0 to Pi/2.
Solution: We know that the formula for the definite integral of sin4x is given by, _{a }∫^{b} sin4x dx = (1/4) (cos 4a  cos 4b). Here, a = 0 and b = π/2. So, we have
_{0} ∫^{π/2} sin4x dx = (1/4) (cos 4(0)  cos 4(π/2))
= (1/4) (cos 0  cos2π)
= (1/4) (1  1)
= 0
Answer: _{0} ∫^{π/2} sin4x dx = 0
FAQs on Integration of Sin 4x
What is Integration of Sin4x in Calculus?
Integration of sin 4x can be calculated using different methods such as the substitution method. The integration of sin 4x is equal to the negative of onefourth of the cosine of the angle 4x plus the constant of integration
What is the Formula for Integration of Sin 4x?
The formula of integration of sin 4x is written as ∫sin 4x dx = (1/4) cos 4x + C, where the mathematical symbols ∫ denote the integration, dx denotes the variable of integration, and C is the constant of integration.
How to Find the Integration of Sin 4x?
The Integration of Sin 4x can be calculated using the substitution method. The general formula for the integral of sin ax is (1/a) cos ax + C.
What is the Integration of Sin 4x by Sin x?
The integration of sin 4x upon sin x is 2 [(1/3) sin 3x + sin x] + C, where C is the constant of integration.
How to Find the Integration of Sin 4x by cos 2x?
The integration of sin 4x upon cos 2x is cos 2x + C.
What is the Definite Integration of Sin 4x From 0 to Pi?
The definite integration of sin 4x from 0 to pi is equal to 0.
What is the Integral of Sin^4x?
The integral of sin^4x is equal to (3/8)x + (1/32) sin(4x) − (1/4) sin(2x) + C, where C is the integration constant. We can evaluate this integral using the power reduction formulas.
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