Indefinite Integration Questions (389)

Prove the reduction formula: $$\int \sec^n x \, dx = \frac{1}{n-1} \sec^{n-2} x \tan x + \frac{n-2}{n-1} \int \sec^{n-2} x \, dx$$

Evaluate: \(\int x \sin x \sec^3 x dx\)

Let f(x) = \int x^2 \cos 2x (2x + 6\tan x - 2x\tan^2 x) dx and f(x) passes through the point (\pi, 0).If f: \mathbb{R} - \{(2n+1)\frac{\pi}{2}\} \to \mathbb{R} then f(x) be a:

Evaluate \(\int e^x \sec x(1 + \tan x) dx\)

Let \( I = \int \dfrac{\left(\sin^{3/2}\theta + \cos^{3/2}\theta\right) d\theta}{\sqrt{\sin^3\theta \cos^3\theta \sin(\theta+\alpha)}} \). Then \(I\) equals:

Find $\int \frac{2^x + 3^{2x+1} + 5^{3x-2}}{7^x} dx$

Evaluate: \ 10\int \cos\left(\frac{x+1}{\sqrt{x^2+2x+5}}\right) dx

[JEE Main 2022] \(\displaystyle\int\frac{\cos^2 x}{(\cos x+\sin x)^3}\,dx\) equals (where \(C\) is a constant)

\(\int e^{6\log x - e^5\log x}}{e^{4\log x - e^3\log x}} dx\) is equal to

Assertion (A): When \(f(x) = \frac{x^2 + 1}{2}\), \(\int \frac{dx}{x} = 2\ln|x| + c\)Reason (R): \(\int (h(x))^n h'(x) dx = \frac{(h(x))^{n+1}}{n+1} + C\)

If \(f(x) = f(x) + xf'(x)\) then \(\int g(x)\,dx\) is equal to:

Let \(n \geq 2\) be a natural number and \(0

Evaluate \(\int \frac{1 + x}{\sqrt{1+x^4}} dx\)

The value of $\displaystyle \int \frac{secx(2+secx)}{(1+2secx)^2} dx$ is equal to

The value of $\displaystyle \int \frac{dx}{(1+x) \sqrt{x} \sqrt{1-x}}$ is equal to

Given \[ I = \int \frac{\tan x}{1 + \tan x + \tan^2 x}\, dx = Kx + A\tan^{-1}\left(\frac{2\tan x + 1}{\sqrt{3}}\right) + C \]Find the values of \(K\) and \(A\).

Evaluate \(\int \frac{f(x)g'(x) - f'(x)g(x)}{f(x)g(x)} \left[\log(g(x)) - \log(f(x))\right]\, dx\)

[JEE Main 2020] \(\displaystyle\int\frac{\sin x+\cos x}{\sqrt{1-\sin 2x}}\,dx\) for \(0

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

∫ (2+√x)dx/x²(√(x+1+√x)) is equal to:

\(\int \frac{\sin 2x}{\cos 7x \sqrt{1 + 2\cos 5x \cos 8x}} dx\) is equal to

If \(\int \frac{dx}{(\sin x + 4)(\sin x - 1)} = A \tan^{-1}\left(\frac{\tan x/2 - 1}{2}\right) + B \tan^{-1} f(x) + C\), then find the value of A.

67. \(\int \frac{1 + \sin x}{\cos x} \, dx\) equals

Evaluate: $\int \frac{dx}{2 + \sin^2 x}$

If $I(x)=\displaystyle\int e^{\sin^2x}(\cos x\sin 2x-\sin x)\,dx$ and $I(0)=1$, then $I\!\left(\dfrac{\pi}{3}\right)$ is equal to

Find $\int \frac{2x^2 + 3x^5}{1+x^6} dx$

[JEE Main 2023] If \(\displaystyle\int x^5 e^{-x^2}\,dx = e^{-x^2}\cdot f(x)+C\) and \(f(0)=-120\), what is the integer value of \(f(1)\)?

Assertion (A): If the primitive of \(f(x) = \sin x + 2x - 4\) has the value 3 for \(x = 1\), then there are exactly two values of \(x\) for which the primitive of \(f(x)\) vanishes.Reason (R): \(\cos x\) has period \(2\pi\).

If \int 2 2x +5x+9 dx = x\sqrtx 2 + x + 1 + \alpha\sqrtx 2 + x + 1+ \beta log ∣ e∣ x + 1 2 + \sqrtx 2 + x + 1∣ ∣ + C , where C is the \sqrtx2 +x+1 constant of integration, then \alpha + 2\beta is equal to _______.

Let \(f(x)\) be a function satisfying \(f'(x) = f(x)\) and \(f(0) = 2\). Then \(\int \frac{f(x)}{3 + 4f(x)} dx\) is

Evaluate $\int \frac{dx}{2x^2+x-1}$

If \(I = \int_{-1}^{1} \frac{\cos^{-1}\left(\frac{x^4}{4}\right)}{1+x^2} dx = k\int_{0}^{1} \frac{\cos^{-1}\left(\frac{x^4}{4}\right)}{1+x^2} dx\), then find \(k\).

\(\displaystyle\int\frac{dx}{(x-\alpha)(x-\beta)}\) equals \((\alpha\neq\beta)\)

Evaluate \(\int \frac{1 - x^2}{x(1 - 2x)} dx\)

Evaluate $\int \frac{dx}{\sqrt{(x-a)(b-x)}} (b > a)$

If $f(x) = \lim_{n \to \infty} \frac{x^n - x^{-n}}{x^n + x^{-n}}$, $0 < x < 1$, $n \in \mathbb{N}$ then $\int (\sin^{-1} x) f'(x) dx$ is equal to:

∫ dx/∛(x⁵(x+1)⁷) is equal to:

Given \(I = \int \cos(\log_e x)\, dx\). Then \(I\) equals:

Let \(I = \int \frac{e^{4x} - e^{2x}}{e^{4x} + e^{2x} + 1} dx\) and \(J = \int \frac{e^{2x}}{e^{4x} + e^{2x} + 1} dx\). Then, for an arbitrary constant C, the value of J – I equals [IIT - 2008]

If $f(x) = \displaystyle\int \frac{1}{x^{1/4}(1+x^{1/4})}\,dx$, $f(0) = -6$, then $f(1)$ is equal to:

Let \(a \in (0, \pi/2)\) be fixed. If the integral \(\int \frac{\tan x - \tan a}{\tan x + \tan a} dx = A(x) \cos 2a + B(x) \sin 2a + C\), where \(C\) is a constant of integration, then the functions \(A(x)\) and \(B(x)\) are respectively

\(\int \frac{1+x\cos x}{x(1+x^2 e^{2\sin x})} dx\) is equal to

Evaluate ∫ (∛x + ∛(2-x²))(∛(1-x²) - ∛(2-x²))dx/∛(1-x³) for x ∈ (0,1):

Evaluate: \(\int \frac{dx}{(x + 2)(x^2 + 1)}\)

If \(\int \frac{\csc^2 x - 2010}{\cos^{2010} x} dx = -\frac{f(x)}{(g(x))^{2010}} + C\); where \(f\left(\frac{\pi}{4}\right) = 1\); then the number of solutions of the equation \(\frac{f(x)}{g(x)} = \{x\}\) in \([0, 2\pi]\) is/are: (where \(\{\}\) represents fractional part function)

If \(\displaystyle\int e^{\sin x}\left(\dfrac{x\cos^3 x - \sin x}{\cos^2 x}\right)dx = e^{\sin x}\big(f(x) - \sec x\big) + C\), then \(\dfrac{f(7)}{2}\) is equal to

Assertion (A): For \(-1 , the value of \((a + k)\) is \(\frac{9}{2}\).Reason (R): The given integral reduces to the form \(\int \frac{f'(x)}{f(x)} dx\) where \(f(x) = (x-1)^{5/2}\).

Let $I(x)=\displaystyle\int\sqrt{\dfrac{x+7}{x}}\,dx$ and $I(9)=12+7\ln 7$. If $I(1)=\alpha+7\ln(1+2\sqrt{2})$, then $\alpha^4$ is equal to _____.

If \int e ( x x sin -1 x + sin -1 x + x ) dx = g(x) + C , where C is the constant of integration, then g ( 1 ) equals : 3/2 1-x 2 2 \sqrt1-x2 (1-x ) 2

Evaluate : $\int \frac{2+3\cos\theta}{\sin\theta+2\cos\theta+3} d\theta$