List of top Mathematics Questions on Integration asked in Maharashtra Class XII

Prove that: \[ \int_0^{2a} f(x) dx = \int_0^a f(x) dx + \int_0^a f(2a - x) dx. \] Hence show that: \[ \int_0^\pi \sin x \,dx = 2 \int_0^{\pi/2} \sin x \,dx. \]
Evaluate: \[ I = \int \frac{5e^x}{(e^x + 1)(e^{2x} + 9)} \, dx. \]
Evaluate: \[ \int \frac{1}{x^2 + 25} dx. \]
Prove that: \[ \int \frac{1}{a^2 - x^2} dx = \frac{1}{2a} \log \left( \frac{a + x}{a - x} \right) + C. \]
Find the area of the region bounded by the curve \( y = x^2 \), and the lines \( x = 1 \), \( x = 2 \), and \( y = 0 \).
Evaluate: \[ \int_{0}^{\frac{\pi}{2}} \cos^2 x \,dx. \]
Evaluate: \[ \int \log x \,dx. \]

If \( \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} x^3 \sin^4 x \, dx = k \), then \( k \) is ____________.