Integration of Sin2x 

The integration of sin2x is a common problem in calculus, especially in indefinite integrals. It appears in various mathematical contexts such as differential equations, wave functions, and physics problems. In this blog, we’ll understand the Integration of Sin 2x Derivation, memorize the Integration of Sin 2x formula, and solve examples to boost your problem-solving skills.

1.0What is the Integration of sin2x?

The integration of sin2x means finding the antiderivative (or the integral) of the trigonometric function sin(2x). This involves using substitution methods or recognizing patterns from standard formulas.

2.0Integration of Sin 2x Derivation

Let’s derive the integral of sin2x step by step:

We know: $\int sin(ax)dx=-\frac{1}{a}cos(ax)+C$

Here, a = 2, so applying the formula:

$\int sin(2x)dx=-\frac{1}{2}cos(2x)+C$

Hence, the Integration of Sin 2x Formula is:

$\int sin(2x)dx=-\frac{1}{2}cos(2x)+C$

3.0Solved Problems on Integration of sin2x

Example 1: $\int sin(2x)dx$

Solution: Using the standard formula,: $\int sin(2x)dx=-\frac{1}{2}cos(2x)+C$

Example 2: $\int 5sin(2x)dx$

Solution: Factor out the constant: $\int sin(2x)dx=5.(-\frac{1}{2}cos(2x))+C$

Example 3: $\int sin(2x+\pi )dx$

Solution: Use trigonometric identity:

$sin(2x+\pi )=sin(2x)=sin(2x+\pi )dx$

$\int -sin(2x)dx=\frac{1}{2}cos(2x)+C$

Example 4: $\int \frac{\sin (2x)}{x}dx$

Solution:

This is a non-elementary integral, i.e., it can't be solved using elementary functions. It is typically expressed using the Sine Integral function (Si).

Hence, the answer is:

$\int \frac{\sin (2x)}{x}dx=\mathrm{Si}(2x)+C$

Example 5: ${\int }_0^{\frac{\pi }{2}}\sin (2x)dx$

Solution:

Use the formula:

$\int \sin (2x)dx=-\frac{1}{2}\cos (2x)$

Now apply limits:

$={\left[-\frac{1}{2}\cos (2x)\right]}_0^{\frac{\pi }{2}}$

$=-\frac{1}{2}\left[\cos (\pi )-\cos (0)\right]$

$=-\frac{1}{2}[-1-1]=1$

4.0Practice Questions on Integration of Sin2x

1. $\int 3\sin (2x)dx$
2. $\int \sin (4x)dx$
3. $\int {\sin }^2(2x)dx$
4. $\int \sin (2x)\cos (2x)dx$
5. $\int x\sin (2x)dx$

5.0Sample Questions on Integration of Sin2x

Q1. What is the integration of sin2x?

Ans: The integration of sin2x is:

$\int \sin (2x)dx=-\frac{1}{2}\cos (2x)+C$

Q2. What formula is used to integrate sin2x?

Ans: We use the standard trigonometric integral:

$\int \sin (ax)dx=-\frac{1}{a}\cos (ax)+C$

For sin2x, a = 2, so the result becomes:

$-\frac{1}{2}\cos (2x)+C$

Q3. Can we use substitution to solve $\int sin(2x)dx$?

Ans:
Yes. Let u = 2x, then $dx=\frac{du}{2}$. So:

$\int \sin (2x)dx=\int \sin (u)\cdot \frac{du}{2}$

$=-\frac{1}{2}\cos (u)+C$

$=-\frac{1}{2}\cos (2x)+C$

Q4. What is the definite integral of sin2x from 0 to $\frac{\pi }{2}$?

Ans:  ${\int }_0^{\frac{\pi }{2}}\sin (2x)dx=1$

Q5. What is the derivative of $-12cos(2x)-\frac{1}{2}\cos (2x)$?

Ans: The derivative of $-\frac{1}{2}cos(2x)$ is:

$\frac{d}{dx}\left(-\frac{1}{2}\cos (2x)\right)=\sin (2x)$

This confirms the correctness of the integral.

Q6. Is the integration of sin2x and sin(x²) the same?

Ans: No.

• $\int \sin (2x)dx$ is solvable using basic trigonometric integration.
• $\int \sin (x^2)dx$ is a non-elementary integral and requires special functions like the Fresnel integral.