31. Evaluating Definite Integrals Using Substitution

Dated: 30-10-2024

Substitution

We can evaluate

\[\int_a^b f(x) dx\]

by following ways

Method 1

First, evaluate the indefinite integral¹ \(\int f(x) dx\) and then use the relationship

\[\int_a^b f(x) dx = \int f(x) dx \bigg]_a^b\]

Method 2

We can write the equation in following form

\[\int_a^b f(x) \, dx = \int_a^b h(g(x)) g'(x) \, dx\]

Then assume \(u = g(x)\), we get

\[\frac{du}{dx} = g^{\prime}(x)\]

\[du = g^{\prime}(x) dx\]

Also, \(u = g(a)\) if \(x = a\) and \(u = g(b)\) if \(x = b\).
Substitute these in our original equation, we get

\[\int_a^b f(x) \, dx = \int_{g(a)}^{g(b)} h(u) du\]

Example

Evaluate

\[\int_0^2 x (x^2 + 1)^3 \, dx\]

Method 1

Assume \(u = (x^2 + 1)\), then \(\frac{du}{dx} = 2x \implies \frac{du}{2x} = dx\)
Substitute these values and we will get

\[\int x(x^2 + 1)^3 \, dx = \frac{1}{2} \int u^3 \, du = \frac{u^4}{8} + C = \frac{(x^2 + 1)^4}{8} + C\]

\[ = \frac{(x^2 + 1)^4}{8} \bigg]_0^2 = 78\]

Method 2

If \(x = 0\) then \(u = 1\)
If \(x = 2\) then \(u = 5\)
Substituting values, we get

\[\int_{0}^{2}x(x^{2}+1)^{3}dx=\frac{1}{2}\int_{1}^{5}u^{3}du=\frac{u^{4}}{8}\bigg]_{1}^{5}=78\]

Approximation by Reimann Sums

The Reimann Sum is the expression

\[\sum_{i=1}^{n} f(x_i^*)\Delta x_i\]

If we take the limit² of this expression, we get a definite integral.³
However, if we don't but \(n\) is relatively pretty large then in that case

\[\int_{a}^{b}f(x)dx\approx\sum_{i=1}^{n}f(x_{i}^{*})\Delta x_{i}\]

References

Read more about notations and symbols.

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1. Read more about integrals. ↩

2. Read more about limits. ↩

3. Read more about definite integral. ↩