Let the slope of the line 45 x + 5 y + 3 = 0 be 27 r 1 + 9 r 2 2 for some r 1 , r 2 ∈ R . Then lim x → 3 ∫ 3 x 8 t 2 3 r 2 x 2 - r 2 x 2 - r 1 x 3 - 3 x d t is equal to ______.

Step-by-step solution

We know that, The slope of line 45 x + 5 y + 3 = 0 will be - 9 Now, given 27 r 1 + 9 r 2 2 = - 9 Now, solving L = lim x → 3 ∫ 3 x 8 t 2 d t 3 r 2 x 2 - r 2 x 2 - r 1 x 3 - 3 x Applying Newton Leibnitz Theorem and L'Hospital Rule we get, ⇒ L = lim x → 3 8 x 2 3 r 2 2 - 2 r 2 x - 3 r 1 x 2 - 3 ⇒ L = 72 3 r 2 2 - 6 r 2 - 27 r 1 - 3 ⇒ L = 72 - 9 r 2 2 - 27 r 1 - 3 Now, using 27 r 1 + 9 r 2 2 = - 9 we get, ⇒ L = 72 9 - 3 = 12