Indefinite Integration

The value of $\int \sin 4x \cos 3x\, dx$ is

$\int \left(e^{\log(\sin x)} + \cos x\right) x\, dx = $

$\int \dfrac{x^4(x^{10} - 1)}{x^{20} + 3x^{10} + 1}\, dx = \cdots$

If $f(x) = \int \dfrac{x}{(x^2 + 4)(x^2 + 9)}\, dx$ and $f(0) = \dfrac{1}{5}\log\left(\dfrac{2}{3}\right)$, then $f(1) = $

$\int(x^{21} + x^6 + x^3)(2x^{18} + 7x^3 + 14)^{\frac{1}{3}}\ dx =$

$\int\dfrac{1}{\sqrt{2x - x^2}}dx =$

$\int\dfrac{\log x}{(1 + \log x)^2}dx =$

If $\int f(x)dx = g(x) + c$ then $\int f^{-1}(x)dx =$

If $f(x), g(x)$ be twice differentiable functions, satisfying $f''(x) = g''(x), f'(1) = 2g'(1) = 4$ and $f(2) = 3g(2) = 9$ then $f(x) - g(x)$ at $x = 4$ is equal to

If $\displaystyle\int \dfrac{dx}{x^{7/2}(x^4 + 1)^{3/8}} = m\left(\dfrac{x^4 + 1}{x^4}\right)^n + c$, where $c$ is a constant of integration, then the value of $\dfrac{n}{m}$ is...

If $f(x) = \dfrac{1}{\log x}$ and $g(x) = \dfrac{1}{(\log x)^2}$, then the value of $\displaystyle\int [f(x) - g(x)]\,dx$ is...

The value of integral $\displaystyle\int \dfrac{\sqrt{x^2 + 1}\,[\log(x^2 + 1) - 2\log x]}{x^4}\,dx$ is equal to...

The value of integral $\displaystyle\int \dfrac{dx}{\sin^2 x + \tan^2 x}$ is...

If $u$ and $v$ are functions of $x$, then $\int \dfrac{1}{v^3}\left(uv\dfrac{du}{dx} - u^2\dfrac{dv}{dx}\right)dx = $

The value of integral $\int x^3 \cos x\,dx$ is...

Let $f(x) = 1 - \dfrac{1}{x}, g_2(x) = f(f(x)), g_3(x) = f(f(f(x)))$ and so on. If $\int x \cdot g_{2026}(x)\,dx = \int g_{2025}(x)\,dx + h(x) + c$, then $h(x) = $...

$\int\dfrac{(x+1)\,dx}{x(1+xe^x)} = \ldots\ldots$

The value of $\int\dfrac{n\sqrt{\text{cosec}^2 x^n - 1}}{x^{(1-n)}}\,dx$ is ...

$\int\sin(\log x)\,dx = $

If $0 \leq x \leq 1$, $I_1 = \int\sin^{-1}\sqrt{1-x^2}\,dx$ and $I_2 = \int\sin^{-1}x\,dx$, then which of the following is true?

Let $f(x) = x$, $f_1(x) = f(\log x)$, $f_2(x) = f_1(\log x)$, $f_3(x) = f_2(\log x)$, $\ldots$ and so on. Then $\int \dfrac{1}{f(x)\,f_1(x)\,f_2(x)\,\ldots f_{2026}(x)}\,dx = \ldots$

If $\int e^{x + \tan^{-1}x}\left(\dfrac{x^2 + 2}{\sec^2(\tan^{-1}x)}\right)dx = e^{f(x)} + c$, then $\ldots$

If $\int f'(x) \cdot e^{x^2}\,dx = (x - 1) \cdot e^{x^2} + k$, where $k$ is constant of integration, then $f(x) = \ldots$

If $f'(x) = \dfrac{(\sqrt{x}+1)e^{\sqrt{x}}}{\sqrt{x}}$ and $f(0) = e$ then $f(1) = \ldots\ldots\ldots\ldots$

$\int x^3 \log x\,dx = $

The value of $\int 2x^{\frac{1}{3}} \sin \sqrt[3]{x^2}\,dx$ is

The value of $\int \dfrac{\sin^3 x}{(\cos^4 x + 3\cos^2 x + 1)\tan^{-1}(\sec x + \cos x)}\,dx$ is

If $f(x) = \dfrac{\sin^{-1}x}{\sqrt{1 - x^2}}$ and $g(x) = e^{\sin^{-1}x}$, then the value of $\int f(x)g(x)\, dx = \ldots$

If $\int\dfrac{3x + 7}{x^2 - 3x + 2}\, dx = m\log\left(\dfrac{x - 2}{x - 1}\right) + n\log(x - 2) + c$, where $m, n \in R$ and $c$ is an integration constant, then $m + n =$

If $\int f(x)\, dx = g(x)$ then $\int x^3 f(x^2)\, dx$ is equal to

If $\int\dfrac{\sin x}{\sin 4x}\, dx = \alpha\log\left|\dfrac{1 + \sin x}{1 - \sin x}\right| + \beta\log\left|\dfrac{1 + \sqrt{2}\sin x}{1 - \sqrt{2}\sin x}\right| + c$, then the value of $32(\alpha + \beta^2) =$

$\displaystyle\int \dfrac{\cos^3 x}{\sin^2 x + \sin x}\,dx =$

$\displaystyle\int \cot^4 x\,dx$ is equal to

$\displaystyle\int \dfrac{(x + 1)(x + \log x)^2}{x}\,dx =$

If $a > 0, b > 0$ and $\int\dfrac{1}{ax^2 + b}dx = \dfrac{1}{\sqrt{6}}\tan^{-1}\left(\dfrac{\sqrt{2}x}{\sqrt{3}}\right) + c$, then $\int\dfrac{1}{bx^2 + a}dx = \ldots$

$\int\dfrac{x^4 + 1}{x^6 + 1}dx = $

If $f(x) = \cos x$, then the value of $\int\dfrac{e^{f(x)}(x\sin^3 x + f(x))}{1 - (f(x))^2}dx = $

If $f\left(\dfrac{x-1}{x+1}\right) = x + 1$, then $\int f(x)\,dx = \cdots$

If an antiderivative of $f(x)$ is $e^x$ and an antiderivative of $g(x)$ is $\cos x$, then $\int f(x)\cos x\, dx + \int g(x)e^x\, dx = $

If $f(x)$ and $g(x)$ are integrable functions then $\left[\int f(x)dx\right]\left[\int g(x)dx\right] = $

$\int \dfrac{\sqrt{x-2}}{x}dx = $

If $\int \sqrt{2}\sqrt{1+\sin x}\, dx = -4\cos(ax+b) + c$, then the value of $a, b$ respectively are...

If $\int \dfrac{1}{\cos^3 x \sqrt{\sin 2x}}\, dx = p(\tan^2 x + q)\sqrt{\tan x} + c$, then the values of $p$ and $q$ respectively are

$\int \text{cosec}^{-1}\left(\sqrt{\dfrac{a+x}{x}}\right)\, \text{d}x =$

The value of $\int \dfrac{\sin^6 x + \cos^6 x}{\sin^2 x \cdot \cos^2 x}\, \text{d}x$ is

$\int \dfrac{x^2\, \text{d}x}{(x^2 + 2)(x^2 + 5)} =$

$\displaystyle\int e^{2x} \dfrac{2(\sin 2x \cos 2x - 1)}{2 \sin^2 2x} \, dx = A \, e^{2x} \cot 2x + c$(Where c is the constant of integration.), then $A^3 =$

If $\displaystyle\int \dfrac{\sin 2x}{\sin^4 x + \cos^4 x} \, dx = f(x) + c$ where c is constant of integration, then the value of $\tan(f(x))$ is

$\displaystyle\int \dfrac{30x^{14} + 15^x \log 225}{x^{15} + 15^x} \, dx =$

$\int \sqrt{1+\sin x}\,dx$ is equal to