What is the integral of (sin^2(x) + cos^2(x)) dx?

What is the integral of (sin^2(x) + cos^2(x)) dx?

• A. x + C
• B. 1/2x + C
• C. 0
• D. Cot(x) + C

Answer: (A)

To determine the integral of \( \sin^2(x) + \cos^2(x) \), it's essential to recognize a fundamental identity from trigonometry. The Pythagorean identity states that \( \sin^2(x) + \cos^2(x) = 1 \) for all values of \( x \). This means that when you set up the integral, it simplifies significantly: \[ \int (\sin^2(x) + \cos^2(x)) \, dx = \int 1 \, dx \] The integral of 1 with respect to \( x \) is simply \( x \), plus a constant of integration \( C \). Therefore, we get: \[ \int 1 \, dx = x + C \] This leads to the conclusion that the integral of \( \sin^2(x) + \cos^2(x) \) is indeed \( x + C \). This is why the correct answer aligns with option A. The other options provide either incorrect expressions that do not correspond to the integral of the constant function 1 or imply different relationships unrelated to the identity.

To determine the integral of ( \sin^2(x) + \cos^2(x) ), it's essential to recognize a fundamental identity from trigonometry. The Pythagorean identity states that ( \sin^2(x) + \cos^2(x) = 1 ) for all values of ( x ).

This means that when you set up the integral, it simplifies significantly:

[

\int (\sin^2(x) + \cos^2(x)) , dx = \int 1 , dx

]

The integral of 1 with respect to ( x ) is simply ( x ), plus a constant of integration ( C ). Therefore, we get:

[

\int 1 , dx = x + C

]

This leads to the conclusion that the integral of ( \sin^2(x) + \cos^2(x) ) is indeed ( x + C ).

This is why the correct answer aligns with option A. The other options provide either incorrect expressions that do not correspond to the integral of the constant function 1 or imply different relationships unrelated to the identity.