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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,

d(sin(mx+b)) by dx = m times cos(mx+b)

Hence

the integral of cos(mx+b)  with respect to x is (1 over m) times sin(mx+b) plus a constant, c

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

the integral of f of (mx+b) with respect to x = (1 over m) times F of (mx+b) plus a constant

Examples

the indefinite integral of cos θ dθ = sin θ + c hence the indefinite integral of cos 2θ dθ = ½sin 2θ + c.

the indefinite integral of e4x+5 dx = ¼e4x+5 + c

the indefinite integral of1 over ((6x minus 10) dx = 1 sixthln |(6x−10)| + c

Exercise

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

Find the indefinite integrals:

indefinite integral =

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

Find the indefinite integrals:

indefinite integral =

<< Multiple, Sum and Difference Rules | Integration Index

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