Integral Calculus
Definite Integrals
MMTS_Full_Test_13
Grade 12
Question:
Let $y=f(x)$ be a differentiable function satisfying $\int_2^x f(t)\,dt+2=\dfrac{x^2}{2}+\int_x^2 t^2 f(t)\,dt$. Then $\int_{-\pi/4}^{\pi/4}\dfrac{f(x)+x^9-x^3+x+1}{\cos^2 x}\,dx=$
Step-by-Step Solution
Key Concept: Differentiate the integral equation to find $f(x)$; then evaluate the definite integral
Differentiating: $f(x)=x+x^2 f(x)\Rightarrow f(x)=\frac{x}{1-x^2}$ (verify). The odd parts vanish; $\int_{-\pi/4}^{\pi/4}\frac{1}{\cos^2 x}dx=2$.
Correct Answer: 1