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Question

Integrate the following functions.
6x+7(x5)(x4)dx


Solution

6x+7(x5)(x4)dx=6x+7x29x+20dx
Let 6x+7=Addx(x29x+20)+B
6x+7=A(2x9)+B6x+7=2Ax+(9A+B)
On equating the coefficients of x and constant term on both sides, we get
2A=6A=3 and 9A+B=7B=34
6x+7x29x+20dx=3(2x9)+34x29x+20dx=32x9x29x+20dx+341x29x+20dx

Let I1=2x9x29x+20dx and I2=1x29x+20dx
6x+7x29x+20dx=3I1+34I2......(i)
Now, I1=2x9x29x+20dx
Let x29x+20=t(2x9)dx=dt
I1=dtt=2t+C1=2x29x+20+C1....(ii)
And I2=1x29x+20dx
x29x+20can be written as x29x+20+814814
Therefore, x29x+20+814814=(x92)214=(x92)2(12)2
I2=1(x92)2(12)2dx=log(x92)+(x92)214+C2

[dxx2a2=log|x+x2a2|]=log(x92)+x29x+20+C2......(iii)
On substituting the values of I1 and I2 from Eqs.(ii)and (iii)in Eq. (i). we get
6x+7x29x+20dx=3[2x29x+20]+34log(x92)+x29x+20+C[3C1+34C2=C]=6x29x+20+34log(x92)+x29x+20+C

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