Mock Test 1 — Definite Integration

Evaluate the trigonometric definite integral $I = \int_0^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx$.

Evaluate the modular absolute value expression integral $\int_{-1}^2 |x^3 - x| \, dx$.

Find the value of the infinite series sum sequence limit: $L = \lim_{n \to \infty} \sum_{r=1}^n \frac{n}{n^2 + r^2}$.

Evaluate the high-power trigonometric integral $\int_0^{\pi/2} \cos^6 x \, dx$ using Wallis's Formula.

Differentiate the definite integral function using Leibniz's Rule: $F(x) = \int_x^{x^2} t^2 \, dt$. Find $F'(2)$.

Evaluate the symmetric interval definite integral $\int_{-\pi/2}^{\pi/2} (\sin^3 x + \cos^2 x) \, dx$.

Evaluate the exponential Type 1 improper integral form: $\int_1^\infty \frac{\ln x}{x^2} \, dx$.

Find the value of the fractional Gamma calculation expression: $\Gamma\left(\frac{5}{2}\right)$.

Evaluate $I = \int_0^\pi \frac{x \sin^3 x}{1 + \cos^2 x} \, dx$ using King's Property transformation.

The sequence limit expression $L = \lim_{n \to \infty} \frac{1}{n^6} \sum_{r=1}^n r^5$ maps directly to which area value?

If $\int_0^{\alpha} \frac{1}{1+4x^2} \, dx = \frac{\pi}{8}$, find the value of the positive spatial limit boundary parameter $\alpha$.

Find the value of the integer parameter $k$ if the definite integral satisfies $\int_0^{50\pi} |\cos x| \, dx = k$.

Evaluate the Gamma exponential improper integral value: $\int_0^\infty x^3 e^{-x} \, dx$.

Assertion (A): The value of the symmetric trigonometric integral $\int_{-1}^1 \sin^5 x \, dx$ is identically equal to 0.
Reason (R): For any function that satisfies the odd reflection identity $f(-x) = -f(x)$, the definite integral across a symmetric interval centered at the origin always vanishes.

Assertion (A): The derivative of the accumulator function $F(x) = \int_0^{x^2} \sqrt{t} \, dt$ is given by $F'(x) = x^2$.
Reason (R): Leibniz's rule requires multiplying the evaluated function term $f(h(x))$ by the derivative of the upper limit boundary function $h'(x)$.