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Found: 3

Problems Calculus Integral Calculating integrals Algebra Trigonometry Identical transformations (trigonometry) Methods Algebraic methods Symmetry and involutions

16–16 4.0

Calculate $\int_{0}^{π / 2} (\sin^{2} (\sin x) + \cos^{2} (\cos x)) d x$ .

Problem #PRU-108604

Problems Geometry Plane geometry Triangles Relationship of side lengths and angles of a triangle. Solving triangles. Ceva's theorem and Menelaus's theorem Calculus Integral Existence of a definite integral Vectors Law of polygon of vectors Similar triangles

13–14 3.0

On the sides $A B$ , $B C$ and $A C$ of the triangle $A B C$ points $P$ , $M$ and $K$ are chosen so that the segments $A M$ , $B K$ and $C P$ intersect at one point and $\vec{A M} + \vec{B K} + \vec{C P} = 0$ Prove that $P$ , $M$ and $K$ are the midpoints of the sides of the triangle $A B C$ .

Problem #PRU-111345

Problems Calculus Integral Applications of integration Area by integration Algebra and arithmetic Polynomials Quadratic polynomials Quadratic equations. Vieta's rule.

16–16 3.0

The numbers $p$ and $q$ are such that the parabolas $y = - 2 x^{2}$ and $y = x^{2} + p x + q$ intersect at two points, bounding a certain figure.

Find the equation of the vertical line dividing the area of this figure in half.