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Indefinite Integrals question

2019 · 10 Apr · Shift 1 · Q38

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JEE Main
Mathematics
Indefinite Integrals
2019 · 10 Apr · Shift 1 · Q38

Indefinite Integrals question

2019 · 10 Apr · Shift 1 · Q38

JEE MainMathematicsIndefinite IntegralsMCQ+4 / −1

If

\int {{{dx} \over {{{\left( {{x^2} - 2x + 10} \right)}^2}}}} = A\left( {{{\tan }^{ - 1}}\left( {{{x - 1} \over 3}} \right) + {{f\left( x \right)} \over {{x^2} - 2x + 10}}} \right) + C

where C is a constant of integration then :

A
A = ${1 \over {54}}$ and f(x) = 9(x–1)2
B
A = ${1 \over {54}}$ and f(x) = 3(x–1)
C
A = ${1 \over {81}}$ and f(x) = 3(x–1)
D
A = ${1 \over {27}}$ and f(x) = 9(x–1)2

View written solutionFree

Correct answer: B

Rewrite the quadratic

We have $x^2-2x+10=(x-1)^2+9.$

So the integral becomes $I=\int \frac{dx}{\left((x-1)^2+9\right)^2}.$

Let $u=x-1 \quad \Rightarrow \quad du=dx.$ Then $I=\int \frac{du}{(u^2+9)^2}.$

Use the standard result

A standard formula is

=\frac{u}{2a^2(u^2+a^2)}+\frac{1}{2a^3}\tan^{-1}\left(\frac{u}{a}\right)+C.$$ Here, $a=3$. Therefore, $$I=\frac{u}{2\cdot 9(u^2+9)}+\frac{1}{2\cdot 27}\tan^{-1}\left(\frac{u}{3}\right)+C.$$ So, $$I=\frac{u}{18(u^2+9)}+\frac{1}{54}\tan^{-1}\left(\frac{u}{3}\right)+C.$$ Substituting $u=x-1$ back, $$I=\frac{x-1}{18\left((x-1)^2+9\right)}+\frac{1}{54}\tan^{-1}\left(\frac{x-1}{3}\right)+C.$$ Since $$((x-1)^2+9)=x^2-2x+10,$$ we get $$I=\frac{1}{54}\tan^{-1}\left(\frac{x-1}{3}\right)+\frac{x-1}{18(x^2-2x+10)}+C.$$ --- 3. **Match with the given form** Given: $$I=A\left(\tan^{-1}\left(\frac{x-1}{3}\right)+\frac{f(x)}{x^2-2x+10}\right)+C.$$ From our result, $$I=\frac{1}{54}\tan^{-1}\left(\frac{x-1}{3}\right)+\frac{x-1}{18(x^2-2x+10)}+C.$$ Factor out $\frac{1}{54}$: $$I=\frac{1}{54}\left(\tan^{-1}\left(\frac{x-1}{3}\right)+\frac{3(x-1)}{x^2-2x+10}\right)+C.$$ Hence, $$A=\frac{1}{54}, \qquad f(x)=3(x-1).$$ --- 4. **Check options** - **A:** $A=\frac{1}{54}$, $f(x)=9(x-1)^2$ ❌ - **B:** $A=\frac{1}{54}$, $f(x)=3(x-1)$ ✅ - **C:** $A=\frac{1}{81}$, $f(x)=3(x-1)$ ❌ - **D:** $A=\frac{1}{27}$, $f(x)=9(x-1)^2$ ❌ So the correct option is **B**. --- 5. **Compare with stored answer** Stored correct answer: **B** Our derived answer: **B** They agree.

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