Find \int \frac{(x+3)}{2x^2+6x+7} dx.

Category: QuestionsFind \int \frac{(x+3)}{2x^2+6x+7} dx.
Editor">Editor Staff asked 11 months ago

Find \int \frac{(x+3)}{2x^2+6x+7} dx.
 
(a) \frac{1}{4} log⁡(2x^2+6x+7) + \frac{3}{4} \left (\frac{1}{\sqrt{2}} tan^{-1}⁡\frac{2x+3}{2\sqrt{2}}\right )+C
 
(b) \frac{1}{4} log⁡(2x^2+6x+7) – \frac{3}{4} (\frac{1}{\sqrt{2}} tan^{-1}⁡\frac{2x+3}{2\sqrt{2}} )+C
 
(c) log⁡(2x^2+6x+7) + \left (tan^{-1}⁡\frac{2x+3}{2√2}\right )+C
 
(d) –log⁡(2x^2+6x+7) – \frac{3}{4} \left (\frac{1}{√2} tan^{-1}⁡\frac{2x+3}{2√2}\right )+C
 
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The question is from Integrals of Some Particular Functions in portion Integrals of Mathematics – Class 12
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Editor">Editor Staff answered 11 months ago

Right answer is (a) \frac{1}{4} log⁡(2x^2+6x+7) + \frac{3}{4} \left (\frac{1}{\sqrt{2}} tan^{-1}⁡\frac{2x+3}{2\sqrt{2}}\right )+C
 
Explanation: We can express
 
x+3=A \frac{d}{dx} (2x^2+6x+7)+B
 
x+3=A(4x+6)+B
 
x+3=4Ax+(6A+B)
 
Comparing the coefficients, we get
 
4A=1 ⇒A=1/4
 
6A+B=3 ⇒B=3/2
 
\int \frac{x+3}{2x^2+6x+7} dx=\frac{1}{4} \int \frac{4x+6}{2x^2+6x+7} dx+\frac{3}{2} \int \frac{1}{2x^2+6x+7} dx
 
Let 2x^2+6x+7=t
 
(4x+6)dx=dt
 
\frac{1}{4} \int \frac{4x+6}{2x^2+6x+7} dx=\frac{1}{4} \int \frac{dt}{t}=\frac{1}{4} log⁡t
 
Replacing t with (2x^2+6x+7)
 
\frac{1}{4} \int \frac{4x+6}{2x^2+6x+7} dx=\frac{1}{4} log⁡(2x^2+6x+7)
 
\frac{3}{2} \int \frac{1}{2x^2+6x+7} dx=\frac{3}{2} \int \frac{1}{2(x^2+3x+\frac{7}{2})} dx=\frac{3}{4} \int \frac{1}{(x+\frac{3}{2})^2+2} dx
 
=\frac{3}{4} \left (\frac{1}{\sqrt{2}} tan^{-1}⁡\frac{2x+3}{2\sqrt{2}} \right )
 
∴\int \frac{x+3}{2x^2+6x+7} dx=\frac{1}{4} log⁡(2x^2+6x+7) + \frac{3}{4} \left (\frac{1}{\sqrt{2}} tan^{-1}⁡\frac{2x+3}{2√2} \right )+C