Indefinite Integration Questions (389)

$\int \frac{dx}{x(x+1)}$ is equal to (where $C$ is an arbitrary constant)

Evaluate $\int \cos \sqrt{x} \, dx$

Find $\int \frac{5\sin x}{\sin x - 2\cos x} dx$

Evaluate $\int \frac{3x+2}{4x^2+4x+5} dx$

If $H(x) = \sin(x^2)$, whose range is $[-1, 1]$. $a = -1, b = 1 \Rightarrow a + 2b = 1$

$(C) \, 2\tan^{-1}(e^x\sin x) + C$

Evaluate \(\int \frac{x+2}{\sqrt{x^2+2x+3}} dx\)

Evaluate \(\int \frac{dx}{9x^2 + 6x + 5}\)

Evaluate \(\int \frac{e^x dx}{\sin x + 1}\)

If ∫ \(\frac{\cos 4x}{\sin^2 x}\) dx = \(A \cot x + B \sin 2x + C\), then find A and B.

Evaluate $\int x^2 e^x \, dx$

\(\int \frac{3\sin x + 2\cos x}{3\cos x + 2\sin x} dx\) is equal to

[JEE Main 2021] \(\displaystyle\int\frac{dx}{x^2(x^4+1)^{3/4}}\) equals (where \(C\) is constant)

86. The anti-derivative of \(f(x) = \log(\log x) + (\log x)^{-2}\) whose graph passes through \((e, e)\), is

If \(xf(x) = 3f^2(x) + 2\), then \(\int \frac{2x - 12xf(x) - f(x)}{(6f(x) - x)(x^2 - f(x))^2} dx\) equals

Evaluate \(\int \frac{\sin 2x}{\sin^4 x} dx\)

[JEE Main 2023] \(\displaystyle\int\frac{3\sin x+2\cos x}{5\sin 2x+3}\,dx\) equals (where \(C\) is constant)

\(\int \sin^2(\ln x) \, dx\) is equal to

Evaluate \[\int \sqrt{\frac{\cos^{\frac{7}{2}} x}{\sin^{\frac{11}{2}} x}}\, dx\]

\(\displaystyle\int\frac{4x^3-7x^2+8x-4}{x^2(1+x^2)}\,dx\) equals (where \(C\) is the constant of integration)

Evaluate: \ 6\int \frac{dx}{x^4 + x^2}

Evaluate \(\int \dfrac{2x+3}{\sqrt{1+x+x^2}}\, dx\).

The integral \(\int \frac{dx}{(x + 4)^{8/7}(x - 3)^{6/7}}\) is equal to (where C is a constant of integration)

Prove that $\int \sqrt{x^2 - a^2} dx = \frac{x}{2} \sqrt{x^2 - a^2} - \frac{a^2}{2} \log |x + \sqrt{x^2 - a^2}| + C$

Let $A_n = \int \tan^n x \, dx$, $\forall n \in \mathbb{N}$. If $A_{n} + A_{n-2} = \frac{\tan^{n-1} x}{n-1} + \lambda$ (where $\lambda$ is an arbitrary constant), then the value of $m$ is equal to

The value of the integral $\int e^{x+1}(2x^2 - \frac{1}{x} + 1) dx$ is equal to (where $C$ is the constant of integration)

If $\int \frac{2\sin x}{\ln(1 + (f(x)) + C}$ (where $x > 0$ and $C$ is the constant of integration) then the range of $f(x)$ is

\(\int e^{x\left(1 + n \cdot x^{n-1} + x^{2n}\right)} \frac{1}{(1+x^n)\sqrt{1+x^{2n}}} dx\) is equal to

Evaluate $\int \frac{dx}{\sqrt{4 - x^2}}$

The value of the integral \(\int \frac{(x)^5}{x^n(1+x^n)^{1/n}} dx\), \(n \in \mathbb{N}\) is

Integral of \(\sqrt{1+2\cot x(\cot x+\csc x)}\) w.r.t. \(x\) is

Integrate \(\int \dfrac{dx}{(3+x^2)\sqrt{1-x}}\).

Evaluate \(\int \sqrt{1 + \sec x}\, dx\)

Evaluate: \ 11\int \frac{\sec^2 x}{(\sec x + \tan x)^n} \, dx, \quad (n > 1)

Evaluate: \ 3\int \frac{\sin^{-1}\sqrt{x}}{\sqrt{1-x}} \, dx

If the integral ∫ 5tanx dx/tanx-2 = x + aln|sinx - 2cosx| + C, then a is equal to:

70. \(\int \frac{x^2}{x^2 + 1} \cdot \frac{x - 1}{x + 1} \, dx\) equals

Find $\int \frac{x^2 - 2x + 3}{\sqrt{x}} dx$

Find $\int \frac{x^3}{\sqrt{x^2+2}} dx$

Evaluate $\int \ln(2x+3)^{(2x+3)} \, dx$

The integral $\displaystyle\int\!\left[\left(\frac{x}{2}\right)^x+\left(\frac{2}{x}\right)^x\right]\log_2 x\,dx$ is equal to

Let $I(x)=\displaystyle\int\dfrac{x+1}{x(1+xe^x)^2}\,dx$, $x>0$. If $\lim_{x\to\infty}I(x)=0$, then $I(1)$ is equal to

$\displaystyle\int_0^\infty\dfrac{6\,dx}{e^{3x}+6e^{2x}+11e^x+6}=$

Evaluate \(\int 2^x (2012)^{\sin^{-1}(2012)} \sin x\, dx\)

If \(f\left(\dfrac{3x-4}{3x+4}\right) = x + 2\), \(x \neq -\dfrac{4}{3}\), and \(\int f(x)\,dx = A\log|1-x| + Bx + C\), then the ordered pair \((A, B)\) is equal to (where \(C\) is a constant of integration)

Let I(x) = \int 11 dx 15 . If I(37) - I(24) = 1 4 ( 1 1 - 1 1 ) , b, c \in N , then 3( b + c) is equal to (x-11) 13 (x+15) 13 b 13 c 13

If f (x) = \int 1/4 1 1/4 dx, f (0) = -6 , then f (1) is equal to : x (1+x )

If $\displaystyle\int\dfrac{\sin^{3/2}x+\cos^{3/2}x}{\sqrt{\sin^3x\cos^3x\sin(x-\theta)}}\,dx=A\sqrt{\cos\theta\tan x-\sin\theta}+B\sqrt{\cos\theta-\sin\theta\cot x}+C$, where $C$ is the integration constant, then $AB$ is equal to

Let $I(x) = \displaystyle\int \frac{dx}{(x-11)^{11/13}(x+15)^{15/13}}$. If $I(37) - I(24) = \dfrac{1}{4}\!\left(\dfrac{1}{b^{1/13}} - \dfrac{1}{c^{1/13}}\right)$, $b, c \in \mathbb{N}$, then $3(b+c)$ is equal to

If $\displaystyle\int \frac{2x^2+5x+9}{\sqrt{x^2+x+1}}\,dx = x\sqrt{x^2+x+1} + \alpha\sqrt{x^2+x+1} + \beta\log_e\!\left|x+\tfrac{1}{2}+\sqrt{x^2+x+1}\right| + C$, where $C$ is the constant of integration, then $\alpha + 2\beta$ is equal to ____.