Integration by substitution

Substitution is the chain rule backwards. It works when an inner function and (up to a factor) its derivative both appear in the integrand.

Rules

Linear substitution: $\int g (a x + b) d x = \frac{1}{a} G (a x + b) + C$
In general: $\int g (h (x)) h^{'} (x) d x = G (h (x)) + C$
Special case: $\int \frac{h ^{'} ( x )}{h ( x )} d x = ln ∣ h (x) ∣ + C$
Check: differentiate the result (chain rule)

Your turn

Three problems on this topic. Type like on a calculator: 3x^2 - 2/x, e^(2x), sqrt(x), ln(x).

Task 1 of 3Substitution

Task

Find an antiderivative:

f (x) = 4 x e^{2 x^{2} + 2}

$F (x) =$ (with or without $+ C$ )

Write like on a calculator: 3x^2 - 2/x, e^(2x), sqrt(x), ln(x), sin(x). Use brackets: e^(2x), not e^2x. Numbers also as 3/4 or e^2 - 1.

Worked example

Find an antiderivative: $f (x) = e^{3 x - 2}$

Worked solution

1.Linear substitution with $a = 3$ : antiderivative of the outer function, divided by $a$ .
2. $F (x) = \frac{1}{3} e^{3 x - 2} + C$

Typical mistake: Multiplying by 3 instead of dividing by 3.

Find an antiderivative: $f (x) = 4 x e^{2 x^{2} + 5}$

Worked solution

1.Substitute $z = 2 x^{2} + 5$ , $d z = 4 x d x$ : the factor 4x is exactly the inner derivative.
2. $\int e^{z} d z = e^{z}$
3. $F (x) = e^{2 x^{2} + 5} + C$

Typical mistake: Missing the derivative in the integrand and starting integration by parts; first check for "inner derivative times outer function".

Common questions

When do I use substitution and when integration by parts?

Substitution when a composite function and its inner derivative appear (e.g. 2x · e^(x²)). Integration by parts when two different kinds of function are multiplied (e.g. x · e^x).

Chain rule Integration by parts Antiderivatives

Integration by substitution

Rules

Your turn

When do I use substitution and when integration by parts?

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