Integral of Sin x – Solved Examples

Example 1: Find the Integral of sin2(x)

Solution:

For sin²(x), you can use the formula involving cos(2x).

∫sin²(x) dx = ∫(1 – cos(2x))/2 dx

Split it into two parts:

= (1/2)∫dx – (1/2)∫cos(2x) dx

Integral of dx is just x. The integral of cos(2x) involves using the sin(2x) formula. It looks like this:

= (1/2)x – (1/4)sin(2x) + C

Combine the two results, and add a constant “C” to account for any potential constant in the original integral.

(1/2)x – (1/4)sin(2x) + C

Example 2: Find the integral of sine³x.

Solution:

Integral of sine cubed with respect to x can be written as:

∫sin³x dx

Use a trigonometric identity to simplify:

sin³x = [1 – cos²(x)] sin(x)

∫[1 – cos²(x)] sin(x) dx

Distribute and separate the terms:

∫[sin x – sin x. cos²(x)]dx

Integrate each term separately:

-cos(x) + 1/3 cos³x + C

Here, ( C ) represents the constant of integration.

Example 3: Find integral of sin x ^-1

Solution:

The integral of sin(x)^-1 can be expressed using the arcsine function. The integral is given by:

∫1/sin x = -ln|cosec x + cot x| + C

Here, (C) is the constant of integration.

Example 4: Find integral of sin x²

Solution:

Integral of sin²(x) with respect to x can be solved using a trigonometric identity.

∫sin2x dx = 1/2∫(1 – cos(2x)dx

Now, integrate each term separately:

1/2​∫(1−cos(2x))dx = 1/2​(∫1dx−∫cos(2x)dx)

= 1/2 [x – 1/2 sin(2x)] + C

where ( C ) is the constant of integration.

Example 5: Find integral of sin x ^-3

Solution:

Integral of sin(x)^-3 with respect to (x) involves a trigonometric substitution. Here’s how you can solve it:

Let u = sin(x), then du = cos(x)dx

Now, substitute these into the integral:

∫sin(x) ⁻³dx = ∫u ⁻³ du

Now, integrate with respect to (u):

∫u⁻³du = u⁻²/−2​ + C

Substitute back in terms of (x) using u = sin(x):

∫sin(x) ⁻³dx = -1/2sin²x + C

So, the integral of sin(x)^-3 with respect to (x) is -1/2sin²x , where (C) is the constant of integration.

Example 6: Find integral of sin inverse x

Solution:

To find the integral of sin^-1(x) with respect to (x), you can use integration by parts. The formula for integration by parts is:

∫udv=uv−∫vdu

u = sin^-1(x) and dv = dx

Now, find (du) and (v):

[Tex]du = \frac{1}{\sqrt{1 – x^2}} \, dx [/Tex]

v = x

Apply the integration by parts formula:

[Tex]\int \sin^{-1}(x) \, dx = x \sin^{-1}(x) – \int x \, \frac{1}{\sqrt{1 – x^2}} \, dx [/Tex]

Now, integrate the remaining term on the right side. You can use substitution by letting (t = 1 – x²), then (dt = -2x , dx):

[Tex]\int x \, \frac{1}{\sqrt{1 – x^2}} \, dx = -\frac{1}{2} \int \frac{1}{\sqrt{t}} \, dt [/Tex]

= √t + C

Now, substitute back in terms of (x):

[Tex] = -\sqrt{1 – x^2} + C [/Tex]

Putting it all together:

[Tex]\int \sin^{-1}(x) \, dx = x \sin^{-1}(x) + \sqrt{1 – x^2} + C [/Tex]

where (C) is the constant of integration.

Example 7: Find integral of x sin 2x dx

Solution:

To find the integral of xsin(2x) with respect to (x), you can use integration by parts. The formula for integration by parts is given by:

∫udv = uv − ∫vdu

u = x and dv = sin(2x)dx

Now, find (du) and (v):

du = dx and v = -1/2cos(2x)

Apply the integration by parts formula:

∫x.sin (2x) dx = −1/2.​x.cos (2x) − ∫−1/2​ cos(2x) dx

Now, integrate the remaining term on the right side. The integral of -1/2cos(2x) can be found by letting (u = 2x) and using a simple substitution:

∫−1/2​cos(2x)dx = −1/4​sin(2x)

Substitute this result back into the original equation:

-1/2x cos(2x) + 1/4 sin(2x) + C

So, the integral of xsin(2x) with respect to (x) is -1/2x cos(2x) + 1/4 sin(2x) + C, where (C) is the constant of integration.

Example 8: Find integral of sin x cos 2x

Solution:

To find the integral of sin(x) cos(2x) with respect to (x), you can use integration by parts. The integration by parts formula is:

∫udv = uv − ∫vdu

u = sin(x) and dv = cos(2x)dx

Now, find (du) and (v):

du = cos(x) dx and v = 1/2 sin(2x)

Apply the integration by parts formula:

∫sin(x).cos(2x)dx = 1​/2sin(x)sin(2x) − ∫1​/2sin(2x)cos(x)dx

Now, integrate the remaining term on the right side. You can use integration by parts again:

∫1/2​sin(2x)cos(x)dx = 1/4​cos(2x)cos(x) − ∫1/4​cos(2x)sin(x)dx

Continue the process until the integral becomes manageable. After simplifying, you will get the final result:

1/2 sin(x)sin(2x) – 1/8 cos(X) cos(2x) + 1/8 sin(X) cos(2x) + C

where (C) is the constant of integration.

Integral of sin x is -cos(x) plus a constant (C). It represents the area under the sine curve. The function repeats every 2π radians due to its periodic nature. This article explains the integral of the sine function, showing its formula, proof, and application in finding specific definite integrals. Further, it mentions solved problems and frequently asked questions.