definite integrals

Question

The value of $\lim_{n 	o \infty} \sum_{r=1}^n \frac{1}{\sqrt{4n^2 - r^2}}$ is:

RankGuru · Accepted Answer

1. Express the limit as a Riemann Sum:
Term $= \frac{1}{\sqrt{4n^2 - r^2}} = \frac{1}{n \sqrt{4 - (r/n)^2}} = \frac{1}{n} \frac{1}{\sqrt{4 - (r/n)^2}}$.

2. Convert to integral:
Let $x = r/n, dx = 1/n$. Limits from $0$ to $1$.
Integral $= \int_0^1 \frac{1}{\sqrt{2^2 - x^2}} dx$.

3. Solve the integral:
Standard formula: $\int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}(x/a)$.
$I = [\sin^{-1}(x/2)]_0^1 = \sin^{-1}(1/2) - \sin^{-1}(0)$.
$I = \pi/6 - 0 = \pi/6$.

Rank Guru

definite integrals

Select the correct option:

More definite integrals Practice Questions

Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $f(x) = x^2 + \int_0^x e^{...

If $I_n = \int_0^{\pi/4} \tan^n x dx$ (for $n > 1$ ), then $I_n + I_{n-2}$ equals:

Evaluate $\int e^x (\frac{1+x\ln x}{x}) dx$ (Assume $x > 0$ and base $e$ for $\ln$ ).

Let $f(x)$ be a continuous function satisfying $f(x)f(a-x) = 1$ . Then $\int_0^a \frac{1}{1 + f(x)} d...

The value of $\int_0^{\pi/2} \frac{\sin^{2025} x}{\sin^{2025} x + \cos^{2025} x} dx$ is:

definite integrals

Select the correct option:

More definite integrals Practice Questions

Let f:R→Rf: \mathbb{R} \to \mathbb{R}f:R→R be a differentiable function such that $f(x) = x^2 + \int_0^x e^{...

If In=∫0π/4tan⁡nxdxI_n = \int_0^{\pi/4} \tan^n x dxIn​=∫0π/4​tannxdx (for n>1n > 1n>1), then In+In−2I_n + I_{n-2}In​+In−2​ equals:

Evaluate ∫ex(1+xln⁡xx)dx\int e^x (\frac{1+x\ln x}{x}) dx∫ex(x1+xlnx​)dx (Assume x>0x > 0x>0 and base eee for ln⁡\lnln).

Let f(x)f(x)f(x) be a continuous function satisfying f(x)f(a−x)=1f(x)f(a-x) = 1f(x)f(a−x)=1. Then $\int_0^a \frac{1}{1 + f(x)} d...

The value of ∫0π/2sin⁡2025xsin⁡2025x+cos⁡2025xdx\int_0^{\pi/2} \frac{\sin^{2025} x}{\sin^{2025} x + \cos^{2025} x} dx∫0π/2​sin2025x+cos2025xsin2025x​dx is:

Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $f(x) = x^2 + \int_0^x e^{...

If $I_n = \int_0^{\pi/4} \tan^n x dx$ (for $n > 1$ ), then $I_n + I_{n-2}$ equals:

Evaluate $\int e^x (\frac{1+x\ln x}{x}) dx$ (Assume $x > 0$ and base $e$ for $\ln$ ).

Let $f(x)$ be a continuous function satisfying $f(x)f(a-x) = 1$ . Then $\int_0^a \frac{1}{1 + f(x)} d...

The value of $\int_0^{\pi/2} \frac{\sin^{2025} x}{\sin^{2025} x + \cos^{2025} x} dx$ is: