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Integration

Linear Substitution

Replacing the variable x in each of the basic functions, such as cos x, by a linear expression mx + b, we get another function, cos(mx+b). By the Chain Rule for differentiation, we see that,

Hence

In general, if F'(x) = f(x) then for any m ≠ 0,

Examples

cos θ dθ = sin θ + c hence cos 2θ dθ = ½sin 2θ + c.

e^4x+5 dx = ¼e^4x+5 + c

dx = ln |(6x−10)| + c

Exercise

Do plenty of exercises until you feel confident with linear substitutions.

Now try some exercises that involve use of the multiple rule too.

<< Multiple, Sum and Difference Rules | Integration Index