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U Substitution Formula

In calculus, u-substitution,is also known as integration by substitution, is a method for finding integrals. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, integration by substitution is an important tool in mathematics. It is the counterpart to the chain rule of differentiation.

This method is intimately related to the chain rule for differentiation. In this method, the function is replaced by U and then integrate according to the fundamental integration formula after integration substitute the real function instead of U.

U substitution formula is given below,

\[\large \int f\left(g\left(x\right)\right){g}’\left(x\right)dx=\int f\left(u\right)du\]

Where,
u = g(x)
du = ${g}’\left(x\right)dx$ 

Solved Example

Question: Evaluate the following integrals: $\int \left(1-\frac{1}{w}\right)cos\left(w-1nw\right)dw$ 

Solution

In this case we know how to integrate just a cosine so let’s make the substitution the stuff that is inside the cosine.

u = w = 1nw

$du=\left(1-\frac{1}{w}\right)dw$

So, as with the first example we worked the stuff in front of the cosine appears exactly in the differential.  The integral is then,

$=\int cos\left(u\right)du$

$=sin\left(u\right)+c$

$=sin\left(w-1nw\right)+c$

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