Question

Solve this question step by step.
Please response it fast .
I need to know that how this is solved.​

Answer 1

Answer:

I = sin⁻¹ (x/5) + c

Step-by-step explanation:

Well, you shouldn't really be worrying about the derivations/proofs of the formulae of this chapter, better try to focus more on learning the formulae and the various manipulations used to solve a question, because eventually if you are able to get a command over these two factors, then at least I can assure you that you can derive these formula within some minutes. This chapter has got nothing to do with Math, it's a completely different world, it's just to enhance your guessing, manipulation, thinking, recalling skills, actually entire integral calculus is based on the same concept, whereas in differential calculus you actually have to think very logically,no fun! But you can enjoy,have fun in integral calculus, honestly.

So,we both know what the result of this integral is,but anyways let's solve it.

So, whenever we have something like √(a² - x² or a² - x²,the very first thing you should do is to take value of x as asinθ and then proceed further. Well, there are many other such handy substitutions which I probably suppose you must be knowing because that's what you are taught in indefinite integration. Told you it's all just some manipulation and substitution, I'm very sure if you might had been aware of this I suppose you wouldn't have asked this question and saved some points xD

According to this question the value of a is 5.

So,x = 5 sinθ ----- (1)

Differentiate it now to obtain :

=> dx = 5 cosθ dθ ---- (2)

Now, substitute the value of x and dx from (1) and (2),

I = ∫ 5 cosθ dθ/ √( 25 - (5 sinθ)²))

I = ∫ 5 cosθ dθ/ √ (25 - 25sin²θ)

Now taking 25 common,

I = ∫ 5 cosθ dθ/ √ [25 (1 - sin²θ)]

I = ∫ 5 cosθ dθ / √(25cos²θ)

I = ∫ 5 cosθ dθ / 5cosθ

I = ∫ dθ

I = θ + c ---- (3)

Now,using (1),

x = 5 sinθ

x/5 = sinθ

∴ θ = sin⁻¹(x/5)

So, our result becomes after substituting value of θ in (3) :

I = sin⁻¹ (x/5) + c