Introduction to integration of sin square x cos square x

In calculus, the integral is a fundamental concept that assigns numbers to functions to define displacement, area, volume, and all those functions that contain a combination of tiny elements. It is categorized into two parts, definite integral and indefinite integral. The process of integration calculates the integrals. This process is defined as finding an antiderivative of a function.

Integrals can handle almost all functions, such as trigonometric, algebraic, exponential, logarithmic, etc. This article will teach you what is integral to a trigonometric function sine. You will also understand how to compute sin integral by using different integration techniques.

What is the integral of sin²(x)cos²(x)?

The integral of sin^2xcos^2x is an antiderivative of sine function which is equal to x/8 – sin(2x)/32. 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^2x formula

The formula of integral of sin contains integral sign, coefficient of integration calculator and the function as sine. It is denoted by ∫(sin²x.cos²x)dx. In mathematical form, the integral of sin^2x cos^2 x is:

$∫\sin^2x.\cos^2xdx=\frac{x}{8}-\frac{\sin(2x)}{32}+c$

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of integral.

How to calculate the integral of sin^2(x)cos^2(x)?

The integral of sin2 xcos2 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:

1. Substitution method
2. Trigonometric formulas
3. Definite integral

Integral of sin2 xcos2 x by using substitution method

The substitution method calculator involves many trigonometric formulas. We can use these formulas to verify the integrals of different trigonometric functions such as sine, cosine, tangent, etc. Let’s understand how to prove the integral of sin square cos square by using the substitution method.

Proof of Integral of sin²xcos²x by using substitution method

To proof the integration of sin square x cos square x by using substitution method, suppose that:

$I=∫\sin^2x.\cos^2xdx$

By using trigonometric identities, we know that,

$\sin2x = 2\sin x.\cos x$

Now, to get substitution of sin^2x.cos^2x, we can use the following identity.

$\sin^2 2x = 4\sin^2x\cos^2x$

The above integral will becomes;

$I=\frac{1}{4}∫\sin^22xdx$

Now we can again simplify the above integral by using another identity. Therefore,

$I=\frac{1}{4}∫(\frac{1–\cos4x}{2})dx$

Then,

$I=∫\frac{1}{8}dx-∫\frac{\cos4x}{8}dx$

Now integrating with respect to x.

$I=\frac{x}{4}–\frac{\sin4x}{32}+c$

Hence we have verified the integral of sin^2x.cos^2x by using a substitution method. This method can also be used to evaluate the integral of sin cos.

Integral of sin^2x.cos^2x by using Trigonometric formulas

The substitution method also involves trigonometric formula calculator that helps to solve integrals easily. Let’s understand how to calculate the integral of sin^2x cos^2x by using different trigonometric formulas.

Proof of integral of sin^2xcos^2x by using Trigonometric formulas

To prove the integral of sin^2xcos^2x, we use different trigonometric formulas. Therefore,

$I=∫\sin^2x\cos^2xdx$

Since we know that:

$\sin^2x=\frac{1 – \cos2x}{2}$

And,

$\cos^2x=\frac{1 + \cos2x}{2}$

Now using these formula in the above integral, we get:

$I=\frac{1}{4}∫(1 – \cos2x)(1 + \cos2x)dx$

More simplification:

$I=\frac{1}{4}∫(1 – \cos^2(2x)dx$

Again we know that

$\cos^2 2x=\frac{1+\cos4x}{2}$

therefore,

$I=\frac{1}{4}∫\left[1 –\left(\frac{1+\cos4x}{2}\right)\right]dx$

Now integrating each term, we get,

$I=\frac{1}{4}\left[\frac{1}{2}–\frac{\cos4x}{2}\right]$

Hence the integral of sin^2xcos^2x is,

$I=\frac{1}{8}–\frac{\cos4x}{8}$

Integration of sin square x cos square x by using definite integral

The definite integral is a type of integral that calculates the area of a curve by using infinitesimal area elements between two points. The definite integral can be written as:

$∫^b_a f(x)dx=F(b) – F(a)$

Let’s understand the verification of the integral of sin x by using the indefinite integral.

Proof of integral of sin2 xcos2 x by using definite integral

To compute the integral of sin^2xcos^2x by using a definite integral, we can use the interval from 0 to π or 0 to π/2. Let’s compute the integral of sin x from 0 to π. For this we can write the integral as:

$∫^π_0 \sin^2x\cos^2xdx=\left|\frac{x}{4}–\frac{\sin4x}{32}\right|^π_0$

Now, substituting the limit in the given function.

$∫^π_0 \sin^2x\cos^2xdx=\frac{π}{4}–\frac{\sin4π}{32} – 0$

Since sin 0 is equal to 0 and sin π is also equal to 0, therefore,

$∫^π_0 \sin^2x\cos^2xdx = \frac{π}{4}$

Which is the calculation of the definite integral of sin xcos^2x. Now to calculate the integral of sin2xcos2x between the interval 0 to π/2, we just have to replace π by π/2. Therefore,

$∫^{\frac{π}{2}}_0 \sin^2x\cos^2xdx=\left|\frac{x}{4}–\frac{\sin4x}{32}\right|^{\frac{π}{2}}_0$

Now,

$∫^{\frac{π}{2}}_0 \sin^2x\cos^2xdx=\frac{π}{8}–\frac{\sin2π}{32} – 0$

Since sin 0 is equal to 0 and sin π/2 is equal to 1, therefore,

$∫^{\frac{π}{2}}_0 \sin^2x\cos^2x dx=\frac{π}{8}$

Therefore, the definite integral of sin2 xcos2 x is equal to π/8.