Introduction to integral of sin^2x*cos x
The integral of sin^2xcos x is a common problem encountered in calculus. It involves finding the antiderivative of the product of sin^2x and cos x with respect to x. This integral has a variety of applications in physics, engineering, and mathematics.
In order to solve this integral, various methods such as substitution, integration by parts, and trigonometric identities can be used. In this article, we will explore the methods of solving the integral of sin^2xcos x, along with its formula, properties, and examples.
What is the integral of sin2(x)cos(x)?
The integral of sin^2xcos x is an antiderivative of sine function which is equal to sin3x/3. It is also known as the reverse derivative of sine function which is a trigonometric identity.
The sine function is the ratio of opposite side to the hypotenuse of a triangle which is written as:
Sin = opposite side / hypotenuse
Integral of sin^2x*cos x formula:
There are different methods to evaluate the integral of sin^2xcos x, such as integration by parts, substitution method, and definite integral method. The formula for the integral of sin^2xcos x using integration by parts is:
∫sin^2x*cos x dx = -1/2 sin^3x + 1/3 sin^3x + C
where C is the constant of integration calculator.
How to calculate the integral of sin2(x)cos(x)?
The integral of sin2xcos x is its antiderivative that can be calculated by using different integration techniques. In this article, we will discuss how to calculate integral of sine by using:
- Integration by parts
- Substitution method
- Definite integral
Integral of sin^2x*cos x by using integration by parts:
In this method, we choose u = sin^2x and dv = cos x dx, and then use integration by parts to evaluate the integral. The formula for this method is:
∫sin^2x*cos x dx = -1/2 sin^3x + 1/3 sin^3x + C
Proof of integral of sin^2x*cos x by using integration by parts:
To prove the integral of sin^2(x)*cos(x) by using integration by parts, we can start by expressing the integral as:
∫sin^2(x) * cos(x) dx
Using the product rule of integration by parts, we can write:
∫sin^2(x) * cos(x) dx = -1/2 ∫sin(2x) dx + C
where C is the constant of integration.
To see why this is true, let u = sin^2(x) and dv = cos(x) dx. Then, du/dx = 2sin(x)cos(x) and v = sin(x).
Using the formula for integration by parts, we have:
∫sin^2(x) * cos(x) dx = uv - ∫vdu
Plugging in the values of u, v, du/dx, and v into this formula, we get:
∫sin^2(x) * cos(x) dx = sin^2(x)*sin(x) - ∫sin(x)*2sin(x)cos(x) dx
Simplifying the first term, we get sin^3(x). For the second term, we can use the identity sin(2x) = 2sin(x)cos(x), to get:
∫sin^2(x) * cos(x) dx = sin^3(x) - ∫sin(2x) dx
Solving this integral, we get:
∫sin^2(x) * cos(x) dx = -1/2 ∫sin(2x) dx + C
Therefore, the integral of sin^2(x)*cos(x) using integration by parts is -1/2 sin(2x) + C.
Integral of sin^2x*cos x by using substitution method:
In this method, we substitute u = sin x and du = cos x dx, and then simplify the integral in terms of u. The formula for this method is:
∫sin^2x*cos x dx = ∫u^2 du
Proof of Integral of sin^2x*cos x by using substitution method:
To prove the integral of sin^2(x)*cos(x) by using the substitution method, we can start by setting u = sin(x), and then finding du/dx.
Using the chain rule, we can find that du/dx = cos(x).
Next, we can express cos(x) in terms of u, using the identity cos^2(x) + sin^2(x) = 1.
Rearranging this identity, we get cos^2(x) = 1 - sin^2(x), so cos(x) = sqrt(1 - sin^2(x)).
Now, we can substitute these expressions into the integral:
∫sin^2(x) * cos(x) dx = ∫u^2 * sqrt(1 - u^2) du
This expression can be solved using a standard technique of integration by substitution.
Let v = 1 - u^2, so that dv/dx = -2u(du/dx).
Rearranging this expression, we get du/dx = -(1/2u) dv/dx.
Substituting u = sin(x), we get du/dx = cos(x) = sqrt(1 - sin^2(x)).
Therefore, we can write dv/dx = -2u(sqrt(1 - sin^2(x)))dx.
Substituting these expressions into the integral, we get:
∫u^2 * sqrt(1 - u^2) du = -1/2 ∫v^(-1/2) dv
Using the power rule of integration, we can solve this integral to get:
-1/2 ∫v^(-1/2) dv = -v^(1/2) + C
Substituting v = 1 - u^2, we get:
-√(1 - u^2) + C
Therefore, the integral of sin^2(x)*cos(x) using the substitution method is -√(1 - sin^2(x)) + C.
Integral of sin^2x*cos x by using definite integral:
In this method, we use the interval from 0 to π/2 for evaluating the integral. The formula for this method is:
∫0π/2sin^2x*cos x dx = ¼
Proof of Integral of sin^2x*cos x:
There are different methods for proving the integral of sin^2x*cos x, such as integration by parts, substitution method, and definite integral method calculator.
Proof of Integral of sin^2x*cos x by using definite integral:
To prove the integral of sin^2x*cos x by using definite integral, we can start by using the double angle identity sin(2x) = 2sin(x)cos(x). We can rewrite the integral as:
∫sin^2(x) * cos(x) dx = ∫(1/2)sin(2x)sin(x) dx
Next, we can apply the product-to-sum identity sin(a)sin(b) = (1/2)(cos(a-b) - cos(a+b)) to get:
∫sin^2(x) * cos(x) dx = (1/4) ∫[cos(x-x) - cos(x+x)] sin(2x) dx
Simplifying this expression, we get:
∫sin^2(x) * cos(x) dx = (1/4) [∫cos(x) sin(2x) dx - ∫cos(3x) sin(2x) dx]
For the first integral, we can use integration by parts by choosing u = sin(2x) and dv = cos(x) dx.
For the second integral, we can use substitution by letting u = 2x and dv = cos(3x) dx. After solving both integrals and simplifying, we get:
∫sin^2(x) * cos(x) dx = (1/4)[(1/2)sin^3(x) - (1/6)sin^3(3x)] + C
This is the definite integral of sin^2x*cos x, where C is the constant of integration.
Applications of the Integral of sin^2(x)*cos(x)
The integral of sin^2(x)*cos(x) is a mathematical expression with a variety of applications in calculus and physics. It can be used to calculate the area under certain curves, simplify trigonometric integrals, and solve differential equations that involve sin^2(x)*cos(x). This integral is also useful in analyzing mechanical systems such as pendulums. For example, the integral can be used to model the motion of a pendulum by relating its position, velocity, and acceleration to the angle of displacement from its rest position.
