Evaluate $\int$(sinx - 5x2)dx

Analyze the question: We need the indefinite integral of sin x − 5x². The "5x2" is read as 5x² (the options with x³ confirm this).

Concept: Integrate term by term, using

• ∫ sin x dx = −cos x
• ∫ xⁿ dx = xⁿ⁺¹/(n+1), for n ≠ −1

Don't forget the constant of integration, k.

Solution:

∫(sin x − 5x²) dx = ∫ sin x dx − 5∫ x² dx

= −cos x − 5·(x³/3) + k

= −cos x − (5x³)/3 + k

Check by differentiating: d/dx[−cos x − 5x³/3] = sin x − 5x², which matches the original integrand.

Why the other options are wrong:

• A and D: they have −10x, which comes from differentiating 5x² instead of integrating it.
• B: it has +cos x, but the integral of sin x is −cos x. (The derivative of cos x is −sin x.)

Answer: C, which is −cos x − 5x³/3 + k.

Tip: Integration raises the power (x² becomes x³/3), while differentiation lowers it. Also remember the sign: the integral of sin is −cos, and the integral of cos is +sin. If you're unsure, differentiate your answer to check.