# What is the antiderivative of #sin (lnx)#?

Thus,

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To do so, we will make use of integration by substitution and integration by parts. We will need to do some additional tricks with these beyond the standard basic methods as well.

Substitution:

Continuing with the substitution,

Integration by parts (i):

Integration by parts (ii):

Again, applying the formula,

Substituting this into the result of the first integration by parts,

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The antiderivative of ( \sin(\ln x) ) is not expressible in terms of elementary functions. It cannot be represented using combinations of basic functions such as polynomials, exponential functions, logarithmic functions, trigonometric functions, and their inverses. However, it can be expressed using special functions like the exponential integral function or the sine integral function, but these representations are more complex and not as commonly used. Therefore, the antiderivative of ( \sin(\ln x) ) is typically denoted by ( \text{Si}(x) ), the sine integral function, defined as:

[ \text{Si}(x) = \int \frac{\sin t}{t} , dt ]

But it's important to note that this integral is not expressible in terms of elementary functions.

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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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