Substitution

# Substitution - THE SUBSTITUTION RULE Example 1 Find 3x2 1...

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THE SUBSTITUTION RULE Example 1. Find Z 3 x 2 p 1 + x 3 dx . Solution Let u = 1+ x 3 . Then, the diﬀerential du = 3 x 2 dx . We can rewrite the indeﬁnite integral as Z u du . Since Z u du = 2 3 u 3 / 2 + C , Z 3 x 2 p 1 + x 3 dx = 2 3 (1 + x 3 ) 3 / 2 + C The technique we used here is called integration by substitution. In general, if we have an integral of the form Z F 0 ( g ( x )) g 0 ( x ) dx, then, by the Chain rule, d dx [ F ( g ( x )] = F 0 ( g ( x )) g 0 ( x ) and Z F 0 ( g ( x )) g 0 ( x ) dx = F ( g ( x )) + C. To make it clear, we usually make the change of variable or substitution u = g ( x ) and get the following rule. The Substitution Rule If u = g ( x ) is a diﬀerentiable function whose range is an interval I and f is continuous on I , then Z f ( g ( x )) g 0 ( x ) dx = Z f ( u ) du Example 2. Find Z cos(4 x ) dx . Solution If we let u = 4 x , then du = 4 dx . Thus, Z cos(4 x ) dx = Z cos u 1 4 du = 1 4 sin u + C = 1 4 sin(4 x ) + C Example 3. Find Z sec xdx . Solution

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Substitution - THE SUBSTITUTION RULE Example 1 Find 3x2 1...

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