How do I evaluate the indefinite integral #int(1+cos(x))^2dx# ?
By signing up, you agree to our Terms of Service and Privacy Policy
To evaluate the indefinite integral [ \int (1+\cos(x))^2 , dx ], use the following steps:

Expand the square: [ (1+\cos(x))^2 = (1+\cos(x))(1+\cos(x)) ] [ = 1 + 2\cos(x) + \cos^2(x) ]

Rewrite (\cos^2(x)) using the trigonometric identity: [ \cos^2(x) = \frac{1 + \cos(2x)}{2} ]

Substitute the expanded expression into the integral: [ \int (1+\cos(x))^2 , dx = \int \left(1 + 2\cos(x) + \frac{1 + \cos(2x)}{2}\right) , dx ]

Integrate each term separately: [ \int 1 , dx = x ] [ \int 2\cos(x) , dx = 2\sin(x) ] [ \int \frac{1 + \cos(2x)}{2} , dx = \frac{1}{2}x + \frac{1}{4}\sin(2x) ]

Combine the results: [ \int (1+\cos(x))^2 , dx = x + 2\sin(x) + \frac{1}{2}x + \frac{1}{4}\sin(2x) ]

Simplify the expression: [ \int (1+\cos(x))^2 , dx = \frac{3}{2}x + 2\sin(x) + \frac{1}{4}\sin(2x) + C ] Where ( C ) is the constant of integration.
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7