Evaluate the integral of 25 x 2 − 4 x \displaystyle \frac{\sqrt{25x^2 - 4}}{x} x 25 x 2 − 4 x x x
Evaluate the definite integral ∫ 0 3 x x 2 + 9 d x \displaystyle \int_{0}^{3} x \sqrt{x^2 + 9} \, dx ∫ 0 3 x x 2 + 9 d x
Solve the integral of ∫ x + 2 3 ( x 2 + 4 x ) ( x + 2 ) − 3 / 2 d x \displaystyle \int \frac{\sqrt{x+2}^3(x^2+4x)}{(x+2)^{-3/2}} \, dx ∫ ( x + 2 ) − 3/2 x + 2 3 ( x 2 + 4 x ) d x
Find the area bounded by the ellipse 2 x 2 + 9 y 2 = 1 2x^2 + 9y^2 = 1 2 x 2 + 9 y 2 = 1
Given the integral with a radical, use the substitution x = 3 sin θ x = 3\sin\theta x = 3 sin θ 9 − x 2 \sqrt{9 - x^2} 9 − x 2
Suppose 4 x 2 + 1 = sec 2 ( θ ) 4x^2 + 1 = \sec^2(\theta) 4 x 2 + 1 = sec 2 ( θ ) x = 1 2 tan ( θ ) x = \frac{1}{2} \tan(\theta) x = 2 1 tan ( θ )
Given the integral ∫ e x e 2 x 9 − e 2 x d x \displaystyle \int \frac{e^x}{e^{2x} \sqrt{9 - e^{2x}}} \, dx ∫ e 2 x 9 − e 2 x e x d x
Evaluate the integral ∫ d x ( x − 3 ) 2 9 − ( x − 3 ) 2
\displaystyle \int \frac{dx}{(x-3)^2 \sqrt{9 - (x-3)^2}}
∫ ( x − 3 ) 2 9 − ( x − 3 ) 2 d x
Using the substitution x = 3 sec ( θ ) x = \sqrt{3} \sec(\theta) x = 3 sec ( θ ) ∫ x 2 − 3 x d x \displaystyle \int \frac{\sqrt{x^2 - 3}}{x} \, dx ∫ x x 2 − 3 d x
Evaluate the integral ∫ 9 − x 2 x 2 d x \displaystyle \int \frac{\sqrt{9-x^2}}{x^2} \, dx ∫ x 2 9 − x 2 d x
Find the area of an ellipse using calculus.
Evaluate the integral ∫ 1 x 2 x 2 + 4 d x \displaystyle \int \frac{1}{x^2 \sqrt{x^2+4}} \, dx ∫ x 2 x 2 + 4 1 d x
Evaluate the integral ∫ 0 3 3 2 x 2 ( 4 x 2 + 9 ) 3 / 2 d x \displaystyle \int_0^{\frac{3\sqrt{3}}{2}} \frac{x^2}{(4x^2 + 9)^{3/2}} \, dx ∫ 0 2 3 3 ( 4 x 2 + 9 ) 3/2 x 2 d x
Using trigonometric substitution, solve the integral ∫ 1 36 − x 2 d x
\displaystyle\int \frac{1}{\sqrt{36-x^2}} \, dx ∫ 36 − x 2 1 d x
Evaluate the integral of 1 x 2 + 6 x \frac{1}{\sqrt{x^2 + 6x}} x 2 + 6 x 1
Evaluate the definite integral ∫ 0 3 2 1 9 − x 2 d x \displaystyle \int_0^{\frac{3}{2}} \frac{1}{\sqrt{9-x^2}} \, dx ∫ 0 2 3 9 − x 2 1 d x
Evaluate the integral ∫ d x x 2 + 2 x + 145 \displaystyle \int \frac{dx}{\sqrt{x^2 + 2x + 145}} ∫ x 2 + 2 x + 145 d x
Using trigonometric substitution, find the integral of 1 x 2 + 4 \frac{1}{\sqrt{x^2 + 4}} x 2 + 4 1 x x x
Evaluate the indefinite integral of x 3 1 − 4 x 2 x^3 \sqrt{1 - 4x^2} x 3 1 − 4 x 2
Identify the type of trigonometric substitution needed (sine, tangent, secant) to integrate a function involving expressions such as a 2 − x 2 a^2 - x^2 a 2 − x 2 a 2 + x 2 a^2 + x^2 a 2 + x 2 x 2 − a 2 x^2 - a^2 x 2 − a 2