Integration by Substitution

Indefinite integral

Let $u = g (x)$ and $F^{'} (x) = f (x)$

\begin{matrix} \int f (g (x)) g^{'} (x) ⅆ x & = & \int f (u) ⅆ u \\ = & F (u) + 𝒸 \\ = & F (g (x)) + 𝒸 \end{matrix}

Riemann integral

Let $u = g (x)$ and $F^{'} (x) = f (x)$

\begin{matrix} \int_{𝒶}^{𝒷} f (g (x)) g^{'} (x) ⅆ x & = & \int_{g (𝒶)}^{g (𝒷)} f (u) ⅆ u \end{matrix}

Symmetries

if $f (- x) = f (x)$ then $\int_{- 𝒶}^{𝒶} f (x) ⅆ x = 2 \int_{0}^{𝒶} f (x) ⅆ x$
if $f (- x) = - f (x)$ then $\int_{- 𝒶}^{𝒶} f (x) ⅆ x = 0$

Examples

\begin{matrix} \int 6 x {(3 x^{2} + 4)}^{4} ⅆ x \\ \int ℯ^{x} \sqrt{ℯ^{x} + 1} ⅆ x \\ \int \frac{3}{x - 10} ⅆ x \\ \int \frac{1}{\sqrt{9 - 4 x^{2}}} ⅆ x \end{matrix}

\begin{matrix} \int_{-1}^{2} x^{2} {(2 x^{3} + 1)}^{5} ⅆ x \\ \int_{1}^{ℯ} \frac{ln x}{x} ⅆ x \\ \int_{1}^{2} \frac{ℯ^{\frac{1}{x}}}{x^{2}} ⅆ x \\ \int_{0}^{\frac{π}{2}} \frac{sin x}{1 + cos x} ⅆ x \end{matrix}

\begin{matrix} \int_{-2}^{2} x^{4} + 3 x^{2} - 7 ⅆ x \\ \int_{0}^{2} x^{4} + 3 x^{2} - 7 ⅆ x \\ \int_{-2}^{2} x^{5} - 8 sin x ⅆ x \end{matrix}

Further examples

Indefinite substitution

\begin{matrix} \int z \sqrt{z^{2} - 5} ⅆ z \\ \int \frac{sin t}{{cos}^{3} t} ⅆ t \\ \int ℯ^{- x} ⅆ x \end{matrix}

Indefinite substitution

\begin{matrix} \int 3 x^{2} ℯ^{2 x^{3}} ⅆ x \\ \int \frac{2 x^{3} + 3 x}{x^{4} + 3 x^{2}} ⅆ x \\ \int \frac{1}{1 + 4 x^{2}} ⅆ x \end{matrix}

Definite substitution

\begin{matrix} \int_{0}^{\frac{π}{2}} {cos}^{2} x ⅆ x \\ \int_{0}^{1} x ℯ^{4 x^{2} + 3} ⅆ x \\ \int_{1}^{2} ℯ^{- x + 1} ⅆ x \end{matrix}

Definite substitution

\begin{matrix} \int_{0}^{1} \frac{1}{\sqrt{1 - x^{2}}} ⅆ x \\ \int_{0}^{\sqrt{3}} \frac{1}{1 + x^{2}} ⅆ x \end{matrix}