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Pattern - 13 of Indefinite Integration

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⋆ Pattern – 13

Integration by parts ∫ uv dx = u ∫ vdx – ∫ $\left(\frac{du}{dx}\ \int v\ dx\right)$ dx

where, L = Logarithm $\rightarrow$ log x, log(2x + 3)

I = Inverse trigo $\rightarrow$ $\sin^{-1}{x},\cos^{-1}{x}.$

A = Algebraic Function $\rightarrow$ $x^2 \times 5x + 4$

T = Trigo $\rightarrow$ sin x, tan x

E = Exponential $\rightarrow$ $e^{2x},\ 5^{7x}$

1 = 1

1) I = ∫ $x^2$ log x dx

Here is known as v

log x is known as u

$=\log{x}\ \int{x^2\ dx}-\ \int\left(\frac{d}{dx}\left(\log{x}\right)\ \int{x^xdx}\right)\ dx$

$=\log{x}\times\frac{x^3}{3}-\ \int\left(\frac{1}{x}\ \times\ \frac{x^3}{3}\right)\ dx$

$=\ \frac{x^3}{3}\log{x}-\frac{1}{3}\ \int x^2 \ dx$

$=\ \frac{x^3}{3}\log{x}-\frac{1}{3}\times\frac{x^3}{3\ }+c$

$=\ \frac{x^3}{3\ }\log{x}-\ \frac{x^3}{9\ }+c$

∫ uv dx = u ∫ v dx – ∫ $\left(\frac{du}{dx}\ \int v\ dx\right) dx$