Let I = ∫ dx/[(x−1)^(11/13)(x+15)^(15/13)]. Then I Is
Solve the integral ∫dx/[(x−1)^(11/13)(x+15)^(15/13)] by substituting t=x−1, factoring out t² from the denominator, and a second substitution to reduce it to a power of a single variable.
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Let f:[1,∞)→R Be a Differentiable Function Such That f(1) = 1/3 and 3∫₁ˣ f(t)dt = x f(x) − x³/3. Then the Value of f(e) Is
Given f(1)=1/3 and 3∫₁ˣf(t)dt=xf(x)−x³/3, find f(e) by differentiating both sides to get a linear differential equation and solving it with an integrating factor.
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Find the Area of the Largest Rectangle That Can Be Inscribed in the Ellipse x²/a² + y²/b² = 1 With Its Sides Parallel to the Coordinate Axes
Find the maximum area of a rectangle inscribed in the ellipse x²/a²+y²/b²=1, using the parametric point (a cosθ, b sinθ) and maximising over θ.
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Find the Area of the Region Bounded by S = {(x, y) : x² + 4x + 2 ≤ y ≤ |x + 2|}
Find the area of the region S={(x,y): x²+4x+2 ≤ y ≤ |x+2|} by substituting u=x+2 to reveal the region’s symmetry and simplify the intersection points.
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For a Positive Constant a, Find dy/dx Where y = a^(t + 1/t) and x = (t + 1/t)^a
Find dy/dx where y=a^(t+1/t) and x=(t+1/t)^a, by treating u=t+1/t as an intermediate variable so the parameter t’s own derivative cancels out.
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The Tangents to the Curve y = (x−2)² − 1 at Its Points of Intersection With the Line x − y = 3 Intersect at the Point
Find where the tangents to y=(x−2)²−1 at its points of intersection with the line x−y=3 meet, by finding the two intersection points, their slopes, and solving the tangent-line system.