List of top Mathematics Questions on Integration

\[ \int \frac{3e^x + 5e^{-x}}{1 - 4e^{-x}} \, dx = 3f(x) + \frac{5}{4}g(x) + \frac{53}{4}\log h(x) + c \] Given the above conditions, find: \[ f(1) + g(1) + h(1) \] 

\(\int \sin^3 2x \sin^{26}x \, dx=\)
\(\int e^{\tan x}(\tan^{7}x+5\tan^{6}x+\tan^{5}x+5\tan^{4}x)\,dx=\)
\(\int \frac{1}{2\cot x-3\tan x}dx =\)
If \([t]\) denotes the greatest integer function, then \[ \int_{-2}^{2} \left[ \frac{x^2+[x+1]} {1+x^2} \right]dx = \ ? \]
The area of the region bounded by the curve \[ y=|x-2|+|x-8|, \] the X-axis and the lines \(x=0\) and \(x=10\) is:
If \[ \int\frac1{1+\cos x}\,dx = \frac1{f\!\left(\frac x2\right)}+C_1, \] then \[ \int f(x)\,dx= \ ? \]
If \[ \int \log x\sqrt{\left(\frac{\log x}{x}\right)^2+\frac1{x^2}}\,dx = \frac{f(x)}{3}\sqrt{1+(\log x)^2}+C \] and \(f(1)=1\), then \(f(e)=\) ?
Evaluate: \[ \int \sin^{-1}x\,dx-\int \cos^{-1}x\,dx. \]
Let \[ f:[0,1]\rightarrow\mathbb R \] be a function defined by \[ f(x)+f(1-x)=1. \] Then \[ \int_0^1 f(x)\,dx= \ ? \]
If \[ \int\frac{dx}{\sqrt{4x^{2}+11x+6}} = \frac12\cosh^{-1}\!\left(\frac{f(x)}{5}\right)+C \] and \[ f(1)=19, \] then \(f(2)=\) ?
Evaluate: \[ \int \frac{\sin^2 x \cos^2 x}{\cos^6 x+\sin^6 x}\,dx \]
If \[ \int (x+5)\sqrt{x-5}\,dx = \frac{2(x-5)^{5/2}}{15}f(x)+c, \] then \(f(6)=\)
Evaluate the integral: \[ \int \frac{x}{x^2-5x+4}\,dx \]
Evaluate: \[ \int \frac{e^{2x}-1}{e^{2x}+e^x+1}\,dx \]
Evaluate: \[ \int \frac{dx}{(x^5+1)^{6/5}} \]
Evaluate: \[ \int \frac{\cos 2x+\sin 4x} {\sqrt{3\sin 2x-2}} \,dx \]
Evaluate: \[ \int_{0}^{\pi} \sqrt{1+4\cos\frac{x}{2}} \left(\cos\frac{x}{2}-1\right)\,dx \]
Evaluate: \[ \int \frac{3x\sec^2\!\sqrt{9x^2-12x+1} -\sec^2\!\sqrt{(3x-2)^2-3}} {\sqrt{9x^2-12x+1}} \,dx \]
Evaluate: \[ \int \frac{1}{\sin x \cos 2x}\,dx \]
Evaluate the definite integral: \[ \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} \frac{x+\frac{\pi}{2}} {2-\sin^2 x} \,dx \]
If \[ \int_{2}^{3} \frac{3\log x} {3\log x+\log(125-75x+15x^2-x^3)} \,dx = k, \] then \[ 4k^2+2k+1= \]
Evaluate: \[ \int \frac{\sin4x}{\sin x}\,dx \]

Water is leaking from the bottom of a conical funnel at the rate of \(0.15\pi\) cm3/s. If the radius of the base is 10 cm and the height is 20 cm, then the rate at which the water level is dropping when it is 5 cm from the top is: 

The integral of \( \int_{-2}^{2} x^4 \, dx \) denominator \( (1+5x^2) \) is: