Find the value of \( \frac{5}{6} + \frac{3}{4} \).

Question

If \[ \int (\sin x)^{-\frac{11}{2}} (\cos x)^{-\frac{5}{2}} \, dx \] is equal to \[ -\frac{p_1}{q_1}(\cot x)^{\frac{9}{2}} -\frac{p_2}{q_2}(\cot x)^{\frac{5}{2}} -\frac{p_3}{q_3}(\cot x)^{\frac{1}{2}} +\frac{p_4}{q_4}(\cot x)^{-\frac{3}{2}} + C, \] where \( p_i, q_i \) are positive integers with \( \gcd(p_i,q_i)=1 \) for \( i=1,2,3,4 \), then the value of \[ \frac{15\,p_1 p_2 p_3 p_4}{q_1 q_2 q_3 q_4} \] is ___________.

Answer 1

To compute \( \frac{5}{6} + \frac{3}{4} \), a common denominator is required for the fractions. The denominators are 6 and 4. The least common multiple (LCM) of 6 and 4 is 12. Both fractions will be converted to this common denominator.

\[ \frac{5}{6} \] is equivalent to \[ \frac{5 \times 2}{6 \times 2} = \frac{10}{12} \].

\[ \frac{3}{4} \] is equivalent to \[ \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \].

The sum of the two fractions is then calculated:

\[ \frac{10}{12} + \frac{9}{12} = \frac{19}{12} \]

The resulting value is \[ \frac{19}{12} \].

Find the value of \( \frac{5}{6} + \frac{3}{4} \).

Show Hint

The Correct Option is C

Solution and Explanation

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