List of top Mathematics Questions on Fundamental Theorem of Calculus

Evaluate: $\int x \, dx$

The derivative of $\sin x$ is:

If $f(x) = x^2 + 3x$, then $f'(x)$ is:

Evaluate the sum: $$ \sum_{n=1}^{\infty} \frac{1}{n(n+1)(n+2)} $$

Let $ \alpha $ and $ \beta $ be the real numbers such that $$ \lim_{x \to 0} \frac{1}{x^3} \left( \frac{\alpha}{2} \int_{0}^{x} \frac{1}{1 - t^2} \, dt + \beta x \cos x \right) = 2. $$ Then the value of $ \alpha + \beta $ is __________.

Evaluate: $$ \lim_{x \to 0} \frac{\sqrt{1 + x} - \sqrt{1 - x}}{x} $$

If $\log_2 (x-1) + \log_2 (x-3) = 3$, find the value(s) of $ x $.

Let \( f : \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \rightarrow \mathbb{R} \) be a differentiable function such that \( f(0) = \frac{1}{2} \). If the \( \lim_{x \to 0} \frac{\int_{0}^{x} f(t) \, dt}{e^{x^2} - 1} = \alpha \), then \( 8\alpha^2 \) is equal to:

Let \(\lim_{n \to \infty} \left( \frac{n}{\sqrt{n^4 + 1}} - \frac{2n}{\left(n^2 + 1\right)\sqrt{n^4 + 1}} + \frac{n}{\sqrt{n^4 + 16}} - \frac{8n}{\left(n^2 + 4\right)\sqrt{n^4 + 16}} + \ldots + \frac{n}{\sqrt{n^4 + n^4}} - \frac{2n \cdot n^2}{\left(n^2 + n^2\right)\sqrt{n^4 + n^4}} \right)\)  be \(\frac{\pi}{k},\) using only the principal values of the inverse trigonometric functions. Then \(k^2\) is equal to ______.

Let \( f(x) = ax^3 + bx^2 + ex + 41 \) be such that \( f(1) = 40 \), \( f'(1) = 2 \) and \( f''(1) = 4 \). Then \( a^2 + b^2 + c^2 \) is equal to:

Let \( f(x) = x^3 + x^2 f'(1) + x f''(2) + f'''(3) \), \( x \in \mathbb{R} \). Then \( f'(10) \) is equal to ______.

Let for a differentiable function \(f : (0, \infty) \rightarrow \mathbb{R}\), \(f(x) - f(y) \geq \log_e \left( \frac{x}{y} \right) + x - y, \quad \forall \; x, y \in (0, \infty).\) 
Then \(\sum_{n=1}^{20} f'\left(\frac{1}{n^2}\right)\) is equal to ____.

Let \([t]\) denote the greatest integer less than or equal to \(t\). Then, the value of the integral \(∫_0^1 [ -8x^2 + 6x - 1] dx\) is equal to