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JEE Main · 2023 · 12 Apr 2023 · Shift 1 · Indefinite Integrals

Let

I (x) = \int \frac{x + 7}{x} dx

and

I (9) = 12 + 7 lo g_{e} 7

. If

I (1) = α + 7 lo g_{e} (1 + 22)

, then

α^{4}

is equal to ______.

Answer: 64

Solution

To evaluate the integral, substitute

x = t^{2}

, so

d x = 2 t d t

. Then:

\int \frac{x + 7}{x} d x = \int 2 t^{2} + 7 d t

Using the standard integral formula

\int t^{2} + a^{2} d t = \frac{t}{2} t^{2} + a^{2} + \frac{a ^{2}}{2} ln ∣ t + t^{2} + a^{2} ∣ + C

, we get:

I (x) = x x + 7 + 7 ln ∣ x + x + 7 ∣ + C

Using the condition

I (9) = 12 + 7 ln 7

:

3 \cdot 4 + 7 ln (3 + 4) + C = 12 + 7 ln 7

This gives

C = 0

.

Therefore:

I (1) = 18 + 7 ln (1 + 8) = 22 + 7 ln (1 + 22)

Comparing with

I (1) = α + 7 lo g_{e} (1 + 22)

, we have

α = 22 = 8

.

Thus

α^{4} = (8)^{4} = 8^{2} = 64

.

JEE Main · 2023 · 12 Apr 2023 · Shift 1 · Integer answer

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