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Definite Integration question

2022 · 28 Jul · Shift 2 · Q29

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JEE Main
Mathematics
Definite Integration
2022 · 28 Jul · Shift 2 · Q29

Definite Integration question

2022 · 28 Jul · Shift 2 · Q29

JEE MainMathematicsDefinite IntegrationMCQ+4 / −1

Let

I_{n}(x)=\int_{0}^{x} \frac{1}{\left(t^{2}+5\right)^{n}} d t, n=1,2,3, \ldots .

Then :

A
$50 I_{6}-9 I_{5}=x I_{5}^{\prime}$
B
$50 I_{6}-11 I_{5}=x I_{5}^{\prime}$
C
$50 I_{6}-9 I_{5}=I_{5}^{\prime}$
D
$50 I_{6}-11 I_{5}=I_{5}^{\prime}$

View written solutionFree

Correct answer: A

We are given $I_n(x)=\int_0^x \frac{1}{(t^2+5)^n}\,dt, \qquad n=1,2,3,\dots$

We need to find a relation involving $I_5$ and $I_6$ .

First, differentiate $I_5(x)$ using the Fundamental Theorem of Calculus: $I_5'(x)=\frac{1}{(x^2+5)^5}.$

Also, $I_6(x)=\int_0^x \frac{1}{(t^2+5)^6}\,dt.$

Observe that $\frac{1}{(t^2+5)^5}=\frac{t^2+5}{(t^2+5)^6}=\frac{t^2}{(t^2+5)^6}+\frac{5}{(t^2+5)^6}.$

So, $I_5(x)=\int_0^x \frac{t^2}{(t^2+5)^6}\,dt+5I_6(x).$

Hence, $\int_0^x \frac{t^2}{(t^2+5)^6}\,dt=I_5-5I_6.$

Now consider $\frac{d}{dt}\left(\frac{t}{(t^2+5)^5}\right).$

Using product rule,

=(t^2+5)^{-5}+t\cdot(-5)(t^2+5)^{-6}(2t).$$ Thus, $$\frac{d}{dt}\left(\frac{t}{(t^2+5)^5}\right)=\frac{1}{(t^2+5)^5}-\frac{10t^2}{(t^2+5)^6}.$$ Now replace $$\frac{1}{(t^2+5)^5}=\frac{t^2+5}{(t^2+5)^6},$$ so $$\frac{d}{dt}\left(\frac{t}{(t^2+5)^5}\right)=\frac{t^2+5-10t^2}{(t^2+5)^6}=\frac{5-9t^2}{(t^2+5)^6}.$$ 5. Integrate both sides from $0$ to $x$: $$\int_0^x \frac{d}{dt}\left(\frac{t}{(t^2+5)^5}\right)dt=\int_0^x \frac{5-9t^2}{(t^2+5)^6}dt.$$ Therefore, $$\frac{x}{(x^2+5)^5}=5I_6-9\int_0^x \frac{t^2}{(t^2+5)^6}dt.$$ Using $$\int_0^x \frac{t^2}{(t^2+5)^6}dt=I_5-5I_6,$$ we get $$\frac{x}{(x^2+5)^5}=5I_6-9(I_5-5I_6).$$ Simplify: $$\frac{x}{(x^2+5)^5}=5I_6-9I_5+45I_6=50I_6-9I_5.$$ 6. Since $$I_5'(x)=\frac{1}{(x^2+5)^5},$$ we have $$xI_5'(x)=\frac{x}{(x^2+5)^5}.$$ Thus, $$50I_6-9I_5=xI_5'.$$ 7. Compare with the options: - A: $50 I_6-9 I_5=x I_5'$ ✅ - B: $50 I_6-11 I_5=x I_5'$ ❌ - C: $50 I_6-9 I_5=I_5'$ ❌ - D: $50 I_6-11 I_5=I_5'$ ❌ Hence the correct option is **A**.

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