How do you find the antiderivative of #3x^2 + sin(4x)+tan x sec x#?
The notation varies based on what you have learned so far, but the idea is to use a substitution.
I'll assume you haven't learned integral notation because you are using the term "antiderivative" instead of "integral".
Assemble the parts and incorporate + C.
You'll learn to refer to this as "integration by substitution" when you study other notation, though it will look slightly different.
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To find the antiderivative of (3x^2 + \sin(4x) + \tan(x)\sec(x)), integrate each term separately:

(\int 3x^2 , dx = x^3 + C_1)

(\int \sin(4x) , dx = \frac{1}{4} \cos(4x) + C_2)

(\int \tan(x)\sec(x) , dx = \int \tan(x) \cdot \frac{1}{\cos(x)} , dx)
Using substitution method:
Let ( u = \cos(x) )
Then, ( du = \sin(x) , dx )
The integral becomes: (\int \frac{1}{u} , du = \lnu + C = \ln\cos(x) + C_3)
Combine all the antiderivatives: [ \int (3x^2 + \sin(4x) + \tan(x)\sec(x)) , dx = x^3  \frac{1}{4} \cos(4x)  \ln\cos(x) + C ]
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To find the antiderivative of the function ( 3x^2 + \sin(4x) + \tan(x) \sec(x) ), integrate each term separately with respect to ( x ) using the rules of integration:
 For ( \int 3x^2 , dx ), use the power rule for integration: ( \int x^n , dx = \frac{x^{n+1}}{n+1} + C ), where ( C ) is the constant of integration.
 For ( \int \sin(4x) , dx ), use the integral of sine function: ( \int \sin(ax) , dx = \frac{1}{a} \cos(ax) + C ), where ( a ) is the coefficient of ( x ) inside the sine function and ( C ) is the constant of integration.
 For ( \int \tan(x) \sec(x) , dx ), use substitution or integration by parts, depending on your preference.
After integrating each term, you will get the antiderivative of the function.
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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.
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