Indefinite Integration Questions (389)

If $\int\left(x^{2010} + x^{804} + x^{402}\right)\left(2x^{1008} + 5x^{402} + 10\right)^{10a}dx = \frac{1}{10a}\left(2x^{2010} + 5x^{804} + 10x^{402}\right)^{10a} + c$, where $c$ is constant then $a$ is equal to

If \(\int \dfrac{3\tan\!\left(x - \dfrac{\pi}{4}\right)}{\cos^2 x\,\sqrt{\tan^3 x + \tan^2 x + \tan x}}\, dx = k\tan^{-1}\!\left(\sqrt{\tan x + 1 + \cot x}\right) + C\), then the value of \(k\) is: [where \(C\) is constant of integration.]

If \(\displaystyle\int x^{26}(x-1)^{17}(5x-3)\, dx = \dfrac{x^{27}(x-1)^{18}}{k} + C\), where \(C\) is constant of integration, then the value of \(k\) is:

If \(\int \frac{dx}{\cos^3 x \cdot \sqrt{2\sin 2x}} = (\tan x)^A + C(\tan x)^B + k\), where \(k\) is a constant of integration, then \(A + B + C\) equals

Let \(I_n = \int \tan^n x\, dx\), \((n > 1)\). If \(I_4 + I_6 = a\tan^5 x + bx^5 + C\), where \(C\) is a constant of integration, then the ordered pair \((a, b)\) is equal to

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

\(\int (g(x) + 1) \sin x\) dx is equal to

If \(I_n = \int \cot^n x \, dx\) (where \(u = \cot x\)), then the value of \(I_2 + I_3 + I_4 + \ldots + I_9 + I_{10}\) is \(l\)

The integral \(\int \sqrt{1 + 2\cot x(\csc x + \cot x)}\, dx\) is equal to (where \(C\) is a constant of integration):

Evaluate \(\int \frac{\cos x + x \sin x}{x(x + \cos x)} dx\)

\(\int \frac{a+b\cos x}{(b+a\cos x)^2} dx\) is equal to

\(\int \frac{x+1}{x\sqrt{x+1}} dx\) is equal to

\(\int \frac{\sin 2x}{\sin^4 x + \cos^4 x} dx = a\cot^{-1}(b\tan 2x) + c\), then

If \(\int f(x)\,dx = g(x)\), then \(\int f(x^{-1})\,dx\) is equal to

Evaluate $\int \sqrt{\frac{3-x}{3+x}} \cdot \sin^{-1}\left(\frac{1}{\sqrt{6}} \sqrt{3-x}\right) dx$

The integral \(\displaystyle\int \dfrac{dx}{(x+1)^{3/4}(x-2)^{5/4}}\) is equal to

We have \[I = \int\left\{\frac{(\log x - 1)}{1 + (\log x)^2}\right\}^2 dx\] Then \(I\) equals:

Let \[I = \int e^{\sin x}\left(\frac{x\cos^3 x - \sin x}{\cos^2 x}\right)dx\]If \(I = e^{\sin x}\cdot f(x) + C\), then \(f(x) = x\) and find the value of \(\dfrac{f(7)}{2}\).

If \(\int \frac{\sqrt{1-x^2}}{x^4} dx = A(x)(\sqrt{1-x^2})^m + C\), for a suitable chosen integer \(m\) and a function \(A(x)\), where \(C\) is a constant of integration, then \((A(x))^m\) equals:

If \(f(x) = \displaystyle\int \dfrac{(3x^4 - 1)}{(x^4 + x + 1)^2}\, dx\) and \(f(0) = 0\), then \(f(-1)\) is equal to:

\(\int \frac{dx}{\cos x - \sin x}\) is equal to

Evaluate: \int \frac{\cos 2x - 2x \operatorname{cosec}^2 2005}{\sin^2 x \cos x} dx

Evaluate: \int x^5 \sqrt{1+x^3} dx

If \(g(1) = g(2)\), then \(\int_1^2 \frac{[f\{g(x)\}]^{-1} f'\{g(x)\} g'(x)}{f^2(x)} dx\) is equal to

Evaluate: \ 5\int \frac{dx}{\sqrt{1+x} - \sqrt[3]{1+x}}

If $y = f(x) = \frac{3x}{2}$ and $g(x) = f^{-1}(x)$, find $g(1)$ where the curve $y = f^{-1}(x)$ passes through $\left(1, -\frac{2}{3}\right)$

The integral \int \frac{(x+1)^3 + x + x^2 dx}{1 + x} is equal to

Find $\int \ln\left(\frac{x+1}{x-1}\right)^2 dx$

Consider the functions f(x) and g(x), both defined from \mathbb{R} \to \mathbb{R}:f(x) = \frac{x^3}{3} + 1 - x \int_{0}^{x} g(t) dtg(x) = x - \int_{0}^{1} f(t) dtThe minimum value of f(x) is:

Evaluate: $\int \frac{dx}{(2 \sin x + 3 \cos x)^2}$

Evaluate $\int e^{ax} \sin bx dx$ and $\int e^{ax} \cos bx dx$

Evaluate $\int \frac{dx}{3x^2+6x+15}$

Evaluate $\int \sin^{-1} x \, dx$

Evaluate $\int e^{\tan x} (\sec x - \sin x) dx$

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

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

Find $\int \sqrt{x^2 + 2x + 5} dx$

Evaluate $\int \sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}} \cdot \frac{1}{x} dx$

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

Find $\int \sqrt{\frac{x-3}{5-x}}dx$

Evaluate $\int \sqrt{x^2 - 3x^6} dx (x > 0)$

Find $\int \frac{2(1+x^2) + 3\sqrt{1-x^2}}{(1+x^2)\sqrt{1-x^2}} dx$

Evaluate the integral: $$I = \int \frac{\tan\left(\frac{\pi}{4} - x\right)}{\cos^2 x \sqrt{\tan^3 x + \tan^2 x + \tan x}} dx$$

Evaluate: $\int \frac{1}{x^4 + 5x^2 + 1} dx$

Find $\int \frac{1 + \sin^3 x + \cos^3 x}{\sin^2 x \cos^2 x} dx$

Evaluate $\int \left( \ln(\ln x^2) + \frac{2}{\ln x^2} \right) dx$

Prove that $\int e^{g(x)} (g'(x) \cdot f(x) + f'(x)) dx = e^{g(x)} \cdot f(x)$.

Evaluate $\int \frac{x(1-x^2)}{1+x^4} dx$

Evaluate $\int \sin^3 x \cos^5 x dx$

Evaluate $\int \frac{x + \sin x}{1 + \cos x} dx$