Integration as inverse of differentiation; indefinite integrals of standard functions; definite integrals as the limit of sums; properties of definite integrals; fundamental theorem of integral calculus; integration by parts, substitution, partial fractions; areas bounded by simple curves. Differential equations — formation, solution of homogeneous DEs of first order and first degree; separation of variables; linear first order DEs.
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Q1 · Integral Calculus · HARD
The value of the integral ∫₀^(π/2) (sin³x)/(sin³x + cos³x) dx is
Q2 · Integral Calculus · MEDIUM
Let f(x) be a continuous function such that ∫₀^x f(t)dt = x² + ∫ₓ^1 tf(t)dt. Then f(1/2) equals