List of top Mathematics Questions on Integration

Evaluate the following integrals: \[ \int \frac{(x^4 + 1)}{x(2x + 1)^2} \, dx \] and \[ \int \frac{1}{x^4 + 5x^2 + 6} \, dx \]

Evaluate the integral: \[ \int \frac{\sqrt{\tan x}}{\sin x \cos x} \, dx \]

The points of extremum of \[ \int_{0}^{x^2} \frac{t^2 - 5t + 4}{2 + e^t} \, dt \] are:

Let
\[ I(R) = \int_0^R e^{-R \sin x} \, dx, \quad R > 0. \]
Which of the following is correct?

If \(\int \frac{\log(x + \sqrt{1 + x^2})}{1 + x^2} \, dx = f(g(x)) + c\), then:

Let \(f : \mathbb{R} \to \mathbb{R}\) be a differentiable function and \(f(1) = 4\). Then the value of
\[ \lim_{x \to 1} \int_{4}^{f(x)} \frac{2t}{x - 1} \, dt \]

If \(f(x) = \frac{e^x}{1+e^x}, I_1 = \int_{-a}^a xg(x(1-x)) \, dx\) and \(I_2 = \int_{-a}^a g(x(1-x)) \, dx\), then the value of \(\frac{I_2}{I_1}\) is:

For any integer \(n\),
\[ \int_{0}^{\pi} e^{\cos^2 x} \cdot \cos^3(2n + 1)x \, dx \text{ has the value.} \]

All values of \(a\) for which the inequality
\[ \frac{1}{\sqrt{a}} \int_{1}^{a} \left( \frac{3}{2} \sqrt{x} + 1 - \frac{1}{\sqrt{x}} \right) dx < 4 \]
is satisfied, lie in the interval.