If then I is equal toa)b)c)d)Correct answer is option 'D'. Can you explain this answer?

Question

If  then I is equal toa)b)c)d)Correct answer is option 'D'. Can you explain this answer?

Answer

Write x^5 as x³. x² and take x³ as t. Now x² will become dt/3.Substitute in x³ and x² and then solve the foll.

Answer

Step 1: Try a substitution

Let v = √(1 + x³)

Then,
v² = 1 + x³ ⇒ x³ = v² - 1

Now differentiate both sides of v² = 1 + x³:
2v dv = 3x² dx ⇒ x² dx = (2v / 3) dv

Now rewrite the integral:

I = ∫ (x³ / √(1 + x³)) dx = ∫ [(v² - 1) / v] * (2v / 3) dv

Simplify:
I = (2 / 3) ∫ (v² - 1) dv

Step 2: Integrate

I = (2 / 3) ∫ (v² - 1) dv = (2 / 3) * [ (v³ / 3) - v ] + C

I = (2 / 9) v³ - (2 / 3) v + C

Now substitute back v = √(1 + x³):

I = (2 / 9) * (1 + x³)^3/2 - (2 / 3) * (1 + x³)^1/2 + C

Final Answer:

I = (2 / 9)(1 + x³)^3/2 - (2 / 3)(1 + x³)^1/2 + C

This matches option d, so the correct answer is:

Option d)