How do you find the antiderivative of #3x+cosx^2#?

Answer 1

Look below

First, divide these

#int3xdx + int cos^2xdx#

Apply the reverse power rule to the first integral.

#\frac{nx^{n+1}}{n+1}#
#int 3x dx = \frac{3x^4}{4}#
now, the #int cos^xdx#
if you remember your trig integrals, you should know #intcos^2xdx = 1/2x+1/4sin2x# Why?

Integration by parts is used.

#intcos^nx=1/ncos^{n-1}xsinx+\frac{n-1}{n}intcos^{n-2}xdx# now to compute it.
#\frac{3x^4}{4}+1/2x+1/4sin2x#
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Answer 2

The antiderivative of ( 3x + \cos(x^2) ) is ( \frac{3}{2}x^2 + C + \int \cos(x^2) , dx ), where ( C ) is the constant of integration. However, the integral of ( \cos(x^2) ) does not have a simple closed-form expression using elementary functions.

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Answer from HIX Tutor

When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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