Indefinite Integration · Mathematics · NDA

$$\int {{{dx} \over {1 + {e^{ - x}}}}} $$ is equal to ?

where, C is the constant of integration.

What is $$\int {{{{x^4} - 1} \over {{x^2}\sqrt {{x^4} + {x^2} + 1} }}dx} $$ equal to ?

What is $$\int {{e^{\sin x}}} {{x{{\cos }^3}x - \sin x} \over {{{\cos }^2}x}}dx$$ equal to ?

What is $$\int {{{\tan }^{ - 1}}(\sec x + \tan x)dx} $$ equal to ?

$$\int {{{(\ln x)}^{ - 1}}dx - \int {{{(\ln x)}^{ - 2}}dx} } $$ is equal to ?

What is $$\int {{{dx} \over {x({x^7} + 1)}}} $$ equal to ?

What is $$\int {{{({x^{e - 1}} + {e^{x - 1}})dx} \over {{x^e} + {e^x}}}} $$ equal to ?

Let f(x) be an indefinite integral of sin²x.

Consider the following statements

Statement I The function f(x) satisfies f(x + $$\pi$$) = f(x) for real x.

Statement II sin²(x + $$\pi$$) = sin²x for all real x.

Which one of the following is correct in respect of the above statements?

What is $$\int {{{\sin }^3}x\cos x\,dx} $$ equal to ?

What is $$\int {{{dx} \over {{a^2}{{\sin }^2}x + {b^2}{{\cos }^2}x}}} $$ equal to ?

What is $$\int {{{dx} \over {{2^x} - 1}}} $$ equal to ?

What is $$\int {{{dx} \over {2{x^2} - 2x + 1}}} $$ equal to ?

What is $$\int {{{dx} \over {x{{(1 + \ln x)}^n}}}} $$ equal to (n $$\ne$$ 1) ?

What is $$\int {\ln ({x^2})dx} $$ equal to?

What is $$\int {{e^{x\ln (a)}}} dx$$ equal to ?

What is $\int (\sin x)^{-1/2} (\cos x)^{-3/2}dx$ equal to?

If $I_1 =\int\frac{e^x dx}{e^x + e^{-x}}$ and $I_2 =\int\frac{dx}{e^{2x} + 1},$then what is I₁ + I₂ equal to?

What is $\rm \int e^{\left(2\ln x+\ln x^2\right)}dx$ equal to?

What is $\rm \int \frac{dx}{\sec x+\tan x}$ equal to?

What is $\int \dfrac{dx}{sec^2({tan}^{-1}x)}$ equal to?