Let f (x) , x ≥ 0 , be a non negative continuous function and let $ F (x) = \int_{0}^{x}f(t)dt $ , x ≥ 0 . If for some c > 0, f (x) ≤ c ….
Q: Let f (x) , x ≥ 0 , be a non negative continuous function and let $ F (x)…
Function f (x) is defined as $ f(x) = x^2 + \int_{-1}^{1} (x^2 + t^2 x^3 )(f (t) – a) dt + a $ , then find the value of a ….
Q: Function f (x) is defined as $\displaystyle f(x) = x^2 + \int_{-1}^{1} (x^2 + t^2 x^3 )(f (t) –…
Evaluate $ \int_{0}^{t} [|sin^{-1}(sinx)|]dx $ ; Where t ∈ ( 2nπ , (8n+1)π/4 )
Q: Evaluate $\displaystyle \int_{0}^{t} [|sin^{-1}(sinx)|]dx $ Where t ∈ ( 2nπ , (8n+1)π/4 ) Solution : Since the function is…
$ \int_{-1/\sqrt{3}}^{1/\sqrt{3}} \frac{cos^{-1}\frac{2x}{1+x^2} + tan^{-1}\frac{2x}{1-x^2}}{e^x +1}$
Q: $\displaystyle \int_{-1/\sqrt{3}}^{1/\sqrt{3}} \frac{cos^{-1}\frac{2x}{1+x^2} + tan^{-1}\frac{2x}{1-x^2}}{e^x +1}$ Solution : Let tan-1x = θ $\displaystyle – 1/\sqrt{3} \le x \le 1/\sqrt{3}$…
Prove that : $ \int_{0}^{\pi/2} \frac{a + b cosx}{(b + acosx)^2} dx = \frac{1}{b}$
Q: Prove that : $\displaystyle \int_{0}^{\pi/2} \frac{a + b cosx}{(b + acosx)^2} dx = \frac{1}{b}$ Solution : We have to…
If roots of the equation x^2 – 10 c x – 11 d = 0 are a, b and those of x^2 – 10 a x – 11 b = 0 are c , d , then find the value of ….
Q: If roots of the equation x2 – 10 c x – 11 d = 0 are a, b and…
If α be a root of the equation 4×2 + 2 x – 1 = 0 , then prove that 4 α 3 – 3α is the other root .
Q: If α be a root of the equation 4 x2 + 2 x – 1 = 0 , then…
If α is a root of equation 3ax^2 + 2bx + 6c = 0 and β is a root of equation –3ax^2 + 2bx + 6c = 0 then prove that ….
Q: If α is a root of equation 3ax2 + 2bx + 6c = 0 and β is a root of…
If each pair of the following three equations x^2 + p1x + q1 = 0, x^2 + p2x + q2 = 0 and x^2 + p3x + q3 = 0 has exactly one root …
Q: If each pair of the following three equations x2 + p1x + q1 = 0 , x2 + p2x…
If ax^2 + 2b x + c = 0 and a1 x^2 + 2 b1 x + c1 = 0 have one and only one root in common, prove that b^2 – ac and b1^2 – a1 c1 must be perfect square.
Q: If ax2 + 2b x + c = 0 and a1 x2 + 2 b1 x + c1 =…
For what integral values of a, the equation x^2 – x(1 – a) – (a + 2) = 0 has integral roots. Find the roots
Q: For what integral values of a, the equation x2 – x(1 – a) – (a + 2) = 0…
Show that the value of tanx/tan3x , whenever defined, never lies between 1/3 and 3.
Q: Show that the value of $\frac{tanx}{tan3x} $ , whenever defined, never lies between 1/3 and 3. Solution : Let…