List of top Mathematics Questions on Definite Integral asked in AP EAMCET

Evaluate the integral \[ \int \sum_{r=0}^{\infty} \frac{x^r 3^r}{2r} dx. \]

Evaluate the integral: \[ \int e^x \left( \frac{x + 2}{(x+4)} \right)^2 dx. \]

Evaluate the integral \[ \int \frac{x^4 + 1}{x^6 + 1} dx. \]

Evaluate the integral \[ \int \frac{x^5}{x^2 + 1} dx. \]

Evaluate the integral \[ \int \frac{\sin^6 x}{\cos^8 x} \, dx. \]

Evaluate the integral $$ I = \int_1^5 \left( |x - 3| + |1 - x| \right) \, dx $$

Evaluate the integral \[ \int_{0}^{\frac{\pi}{4}} \frac{x^2}{(x \sin x + \cos x)^2} dx. \]

Evaluate the integral \[ I = \int_{-1}^{1} \left( \sqrt{1 + x + x^2} - \sqrt{1 - x + x^2} \right) \, dx \]

Evaluate the integral \[ \int e^x (x+1)^2 dx. \]

Evaluate the integral \[ I = \int_0^1 \frac{x}{(1 - x)^{3/4}} \, dx \]

The value of
\[ \int_0^1 a^k x^k\,dx \] is

Let \(\alpha\) and \(\beta\), \((\alpha\lt \beta)\), are roots of \(18x^2-9\pi x+\pi^2=0\), \(f(x)=x^2\), \(g(x)=\cos x\). Then \(\displaystyle \int_{\alpha}^{\beta} x(gof(x))dx=\)

\[ \int_{0}^{\pi} x\left(\sin^2(\sin x)+\cos^2(\cos x)\right)dx= \]

\[ \lim_{n\to\infty} \left( \frac{1}{1+n^5} + \frac{2^4}{2^5+n^5} + \frac{3^4}{3^5+n^5} +\cdots+ \frac{n^4}{n^5+n^5} \right) = \]

If $\int \cos x . \cos 2x . \cos 5x dx = A \; \sin 2x + B \sin 4x + C \sin 6x + D \sin 8x + k $ (where $k$ is the arbitrary constant of integration), then $\frac{1}{B} + \frac{1}{C} = $