Test: Integration By Substitution - JEE MCQ

cos(sin^-1x)/(1-x²)^½……………….(1)
t = sin^-1 x
dt = dx/(1-x²)^½
Put the value of dt in eq(1)
= ∫cost dt
= sint + c
= sin(sin^-1 x) + c
⇒ x + c

So Looking at this integral, we have



 

xe^x = t
(xe^x + e^x) dx = dt
= e^x(x + 1) dx = dt
= ∫dt/sin^2t
= ∫cosec^2t dt
= -cot(xe^x) + c

Create ((X+B) + (A-B)) in numerator and then apply Sin(a+b) formula then you will be able to solve it.

Velocity v(t) = v(0) + ∫₀ᵗ a(τ) dτ = 40 + ∫₀ᵗ (τ² + τ) dτ.

v(t) = 40 + [τ³/3 + τ²/2]₀^t = 40 + t³/3 + t²/2.

Distance s = ∫₀¹⁰ v(t) dt = ∫₀¹⁰ (40 + t³/3 + t²/2) dt.

s = [40t + t⁴/12 + t³/6]₀¹⁰.

s = 40×10 + 10⁴/12 + 10³/6 = 400 + 10000/12 + 1000/6.

10000/12 + 1000/6 = 10000/12 + 2000/12 = 12000/12 = 1000, so s = 400 + 1000 = 1400 km, which corresponds to option C.