Verified answer

Question :- Find the integration of the following :-

[tex]\rm \: \displaystyle\int \:\frac{ {x}^{3} + 3 }{ {x}^{2} - 7x + 12} \: dx\\ \\ [/tex]

[tex]\large\underline{\sf{Solution-}}[/tex]

Given integral is

[tex]\rm \: \displaystyle\int \:\frac{ {x}^{3} + 3 }{ {x}^{2} - 7x + 12} \: dx\\ \\ [/tex]

[tex]\rm \: = \: \displaystyle\int \:\frac{ {x}^{3} + 3 }{(x - 4)(x - 3)} \: dx \\ \\ [/tex]

Now, using Partial Fraction, we have

[tex]\rm\:\frac{ {x}^{3} + 3 }{(x - 4)(x - 3)} = x + 7 + \frac{a}{x - 4} + \frac{b}{x - 3} - - - (1) \\ \\ [/tex]

[tex]\rm \: {x}^{3} + 3 = (x + 7)(x - 4)(x - 3) + a(x - 3) + b(x - 4) \\ \\ [/tex]

On substituting x = 3, we get

[tex]\rm \: {3}^{3} + 3 = 0 + 0 + b(3 - 4) \\ \\ [/tex]

[tex]\rm \: 27 + 3 = - b \\ \\ [/tex]

[tex]\bf\implies \:b \: = \: - \: 30 \\ \\ [/tex]

On substituting x = 4, we get

[tex]\rm \: {4}^{3} + 3 = 0 + a(4 - 3) \\ \\ [/tex]

[tex]\rm \: 67 = a(1) \\ \\ [/tex]

[tex]\bf\implies \:a \: = \: 67 \\ \\ [/tex]

On substituting the values of a and b, in equation (1), we get

[tex]\rm\:\frac{ {x}^{3} + 3 }{(x - 4)(x - 3)} = x + 7 + \frac{67}{x - 4} - \frac{40}{x - 3} \\ \\ [/tex]

On integrating both sides w. r. t. x, we get

[tex]\rm\:\displaystyle\int \:\frac{ {x}^{3} + 3 }{(x - 4)(x - 3)}dx = \displaystyle\int \:(x + 7)dx + \displaystyle\int \frac{67}{x - 4}dx - \displaystyle\int \frac{40}{x - 3}dx \\ \\ [/tex]

[tex]\rm\:\displaystyle\int \:\frac{ {x}^{3} + 3 }{(x - 4)(x - 3)}dx = \frac{ {x}^{2} }{2} + 7x + 67log |x - 4| - log |x - 3| + c \\ \\ [/tex]

[tex]\rule{190pt}{2pt} \\ [/tex]

[tex] { \red{ \mathfrak{Additional\:Information}}}[/tex]

[tex]\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \displaystyle \int \rm \:f(x) \: dx\\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf kx + c \\ \\ \sf sinx & \sf - \: cosx+ c \\ \\ \sf cosx & \sf \: sinx + c\\ \\ \sf {sec}^{2} x & \sf tanx + c\\ \\ \sf {cosec}^{2}x & \sf - cotx+ c \\ \\ \sf secx \: tanx & \sf secx + c\\ \\ \sf cosecx \: cotx& \sf - \: cosecx + c\\ \\ \sf tanx & \sf logsecx + c\\ \\ \sf \dfrac{1}{x} & \sf logx+ c\\ \\ \sf {e}^{x} & \sf {e}^{x} + c\end{array}} \\ \end{gathered}\end{gathered}[/tex]