List of top Mathematics Questions on Integration asked in AP EAPCET

If \(f(x)\) is anti-derivative of \(g(x)\) and \[ \int f(x)g(x)(1+f^2(x))\,dx=F(x), \] then \(F(x)=\)

If \(k\in \mathbb{N}\), then \[ \int \frac{1-k\cos^2 x}{\sin^k x\cos^2 x}\,dx= \]

If \[ \int e^x\{f(x)-f'(x)\}=g(x)+C, \] then \[ \int e^x f'(x)\,dx= \]

If \(a,b,c,d\neq 0\) and \[ f\left(\frac{at+b}{ct+d}\right)=t, \] then \[ \int f(x)\,dx= \] is:

If \[ \int \frac{3x+4}{x^3-2x+4}\,dx=\log f(x)+C, \] then \(f(3)=\)

If \[ \int \frac{e^{\sqrt{x}}}{\sqrt{x}}(x+\sqrt{x})\,dx = e^{\sqrt{x}}\left[Ax+B\sqrt{x}+C\right]+K, \] then \(A+B+C=\)

If \[ \int \frac{1+\tan x}{\sin 2x}\,dx=A\log \tan x+B\tan x+C, \] then \(4A-2B=\)

Evaluate \[ \int \frac{1+\tan x\tan(x+a)}{\tan x\tan(x+a)}\,dx \]

If \[ f(x)=\int x^2\cos^2x\left(2x\tan^2x-2x-6\tan x\right)\,dx \] and \(f(0)=\pi\), then \(f(x)=\)

Evaluate the integral \( \int \frac{x^2 - 2}{x^3 \sqrt{x^2 - 1}} \, dx \)

Let $$ I_n = \int (\cos^n x + \sin^n x) \, dx,\quad \text{and} \quad \frac{n - 1}{n} I_{n - 2} = \frac{\sin x \cos x}{n} f(x) $$ Then, what is $ f(x) $?

\([ \cdot ]\) represents the greatest integer function. If \( \int_{\sqrt{\frac{18}{3}}}^{[x]} dx = a + b\sqrt{2} + c\sqrt{3} \), then find the value of \( a + b + c \).

Evaluate: $$ \int \frac{e^{\tan^{-1}x}}{1+x^2} \left[\left(\sec^{-1}(\sqrt{1+x^2})\right)^2 + \cos^{-1} \left( \frac{1 - x^2}{1 + x^2} \right)\right] dx $$

Evaluate the integral: \[ \int_{-\pi}^{\pi} \frac{2x(1 + \sin x)}{1 + \cos^2 x}\, dx \]

Evaluate: $$ \int \frac{dx}{(x - 3)^{4/5} (x + 1)^{6/5}} = ? $$

Let $ T>0 $ be a fixed number. If $ f: \mathbb{R} \to \mathbb{R} $ is a continuous function such that $ f(x + T) = f(x) $, then: $$ \text{If } I = \int_0^T f(x)\, dx,\quad \text{then } \int_0^{5T} f(2x)\, dx = ? $$

If $$ \int_1^3 x^n \sqrt{x^2 - 1} \, dx = 6 $$ Then find the value of $ n $.

Given: $$ \int \frac{3x + 4}{x^3 - 2x + 4} \, dx = \log f(x) + C $$ Then: $$ f(3) = ? $$

Let \( f(x) = \int \frac{e^{3x}}{4 + 8e^{2x} + e^{4x}} \, dx \), and \( g(x) = \int \frac{2\, dx}{e^{3x} + 8e^x + 4e^{-x}} \), then \( f(x) - g(x) = \) ?

If \( a, b, c, d \ne 0 \) and \( \int \frac{at + b}{ct + d} dt = t \), then find \( \int f(x) dx \) where \( f(x) = \frac{ax + b}{cx + d} \)

Evaluate: \( \int_{0}^{4} \frac{x + 2}{\sqrt{4x - x^2}}\, dx \)

Evaluate \( \int \frac{dx}{1 + a \cos x} \), given \( a>|\sec \theta| \)

The integral $ \int (3x^2 - 2x + 1) \, dx $ is: