Question

∫(1-сosx)²dx helpppppppppppppppppppppppppppppp

Answer 1

$\int\limits {(1-cosx)^2} \, dx\\1-cosx=a\\\int\limits {a^2} \, dx= \frac{a^3}{3}=\frac{(1-cosx)^3}{3}$

Answer 2

Sure, I'd be happy to help you with the integration!

To integrate ∫(1 - cos(x))² dx, we can use trigonometric identities to simplify the expression before integrating.

Step 1: Expand the square term. (1 - cos(x))² = (1 - 2*cos(x) + cos²(x))

Step 2: Apply the trigonometric identity. cos²(x) = (1 + cos(2x))/2

Now the expression becomes: (1 - cos(x))² = (1 - 2*cos(x) + (1 + cos(2x))/2)

Step 3: Distribute the terms inside the square. (1 - cos(x))² = 1 - 2cos(x) + 1/2 + cos(2x)/2 = 3/2 - 2cos(x) + cos(2x)/2

Now we can integrate the expression term by term.

∫(1 - cos(x))² dx = ∫(3/2 - 2*cos(x) + cos(2x)/2) dx

Step 4: Integrate term by term.

∫(3/2) dx = (3/2) * x + C₁ (where C₁ is the constant of integration)

∫(-2*cos(x)) dx = -2 * ∫cos(x) dx = -2 * sin(x) + C₂ (where C₂ is the constant of integration)

∫(cos(2x)/2) dx = (1/2) * ∫cos(2x) dx = (1/2) * (1/2) * sin(2x) + C₃ (where C₃ is the constant of integration)

Step 5: Combine the results.

∫(1 - cos(x))² dx = (3/2) * x - 2 * sin(x) + (1/4) * sin(2x) + C

where C = C₁ + C₂ + C₃ is the constant of integration for the whole expression.

So the final result of the integral is: ∫(1 - cos(x))² dx = (3/2) * x - 2 * sin(x) + (1/4) * sin(2x) + C