Maths · Fundamental integrals involving algebraic, trigonometric, exponential and logarithmic functions
Assertion If and then the value of the integral will be of the type + \) where a
Assertion If \( a>0 \) and \( b^{2}-4 a c<0 . \) then the value of the integral \( \int \frac{d x}{a x^{2}+b x+c} \) will be of the type \( \mu \tan ^{-1}\left(\frac{x+A}{B}\right)+ \) \( C ; \) where \( A, B, C, \mu \) are constant. Reason \( f\left(a>0, b^{2}-4 a c<0, \text { then } a x^{2}+b x+\right. \) \( c \) can be written as sum of two squares.
- A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
- B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
- C. Assertion is correct but Reason is incorrect
- D. Both Assertion and Reason are incorrect
Step-by-step solution
Given a>0 and discriminant negative, the quadratic can be expressed as sum of two squares (completing the square), leading to integral of form ∫ dx/( (x+p)^2 + q^2 ) = (1/q) arctan((x+p)/q) + C, which matches μ arctan((x+A)/B)+C. Thus both statements are correct and reason correctly explains assertion.