Indefinite integration practice · Easy

Option B is correct because the integral is solved by manipulating the integrand using trigonometric identities and then integrating.

Option D ($x{e^{x + {x^{ - 1}}}} + \,c$) is correct because it is the result of integrating the given expression using integration by parts.

The correct answer is B because after substituting $2x+1=t$ and integrating, we obtain the given expression.

Option B ($\int \limits_a^b \,f(x)\,dx$) is correct because after substitution and simplification, the integral transforms to this form.

Option A is correct because by substituting $(p+q)x$ into the antiderivative $f(x)$, we obtain $f((p+q)x)$.

Option C ($e^{\tan ^{-1} x} \cdot\left(\sec ^{-1}\left(\sqrt{1+x^2}\right)\right)^2+C$) is correct because after simplification and substitution, the integral can be solved to obtain this result.

Option B ($\sec ^{-1}(2 x-7)+C$) is correct because the integral can be simplified to a standard inverse secant form by completing the square.

Option C ($2 \tan ^{-1} \sqrt{\frac{x^2+x+1}{x}}+C$) is correct because after algebraic manipulation and substitution, the integral simplifies to this inverse tangent form.

Option A ($\frac{1}{{\log \,a}}\,{\sin ^{ - 1}}\,\left( {{a^x}} \right)\, + \,c$) is correct because it is the result of integrating the given expression using substitution.

Option D, $\log \,\left[ {\log \,\left( {\cosh \,x} \right)} \right] + c$, is correct because it results from a u-substitution where $u = \log(\cosh x)$.

Option A (${e^x}{\sin ^{ - 1}}x + c$) is correct because the integral can be simplified using integration by parts, resulting in this expression.

Option B ($x \sin (\log _e x)+C$) is correct because the integral of the given expression simplifies to this form using substitution and a standard integral formula.

Option A is correct because the assertion is true, and the reason correctly explains why the assertion is true. The definite integral's value depends on the function and interval, not the variable used for integration.

Option A is correct because the integral evaluates to $-\frac{\sqrt{1+x^2}}{x} + c$ after applying the trigonometric substitution.

Option D is correct because the integral can be solved by using a simple substitution.

Option A (a = e, b = -2) is correct because by substituting log x = z and integrating by parts, we can find the values of a and b to be e and -2, respectively.

Option B (13 tan ^{- 1}( \fracx , + ,23 ) + ,\c$\frac{1}{3}\\\,{\tan ^{ - 1}}\left( {\frac{{x\\,\\ + \\,2}}{3}} \right)\\ + \\,\c$) is correct because it is the result of integrating the given expression by completing the square.

Option D ($x\,\,\log \,2x\, - \,x\, + \,c\;$) is correct because it is the result of integrating $\log 2x$ using integration by parts.

Option C is correct because after integrating x e^(2x) dx using integration by parts, the resulting expression matches the form e^(2x)f(x) + C, where f(x) = (2x-1)/4.

Option B ($\frac{\sin \sqrt{x}}{\sqrt{x}}$) is correct because differentiating $-2 \cos \sqrt{x}+C$ gives $\frac{\sin \sqrt{x}}{\sqrt{x}}$.

Option A ($x{(\log \\,x)^n}$) is correct because integrating by parts simplifies the expression to $x{(\log \\,x)^n}$.

Option C ($-e^x \cot x+c$) is correct because the integral simplifies to the form $\int e^x [f(x) + f'(x)] dx = e^x f(x) + c$, where $f(x) = -\cot x$ and $f'(x) = \csc^2 x$.

Option C ($\sin ^{-1}(\log x)+c$) is correct because the integral simplifies to this form after substitution.

Option A is correct because the integral can be rewritten in the form $\int e^x[f(x) + f'(x)] dx$, where $f(x) = \log(\sec x)$ and $f'(x) = \tan x$.

Option B ($3\,\,{\left( {1 + {x^{ - 1/2}}} \right)^{ - 2/3}} + C$) is correct because it is the result of the integration after suitable substitution and simplification.

Option B ($\log \frac{1}{1+e^x}$) is correct because it is the result of evaluating the integral and expressing it in the required form.

Option D (\log_e(1 + \log_e x) + C\\) is correct. The integral simplifies after applying the change of base formula for logarithms and using substitution.

Option C ($2\sqrt {\tan \\,x}$) is correct because the integral of the given expression simplifies to $2\sqrt{\tan x} + c$ after substitution.

Option D (1) is correct because integrating the function y = (3/2)sqrt(x) from x = 0 to x = 1 yields an area of 1.

Option C ($- \frac{1}{2}\,{\left( {{e^{2x}} + \,1} \right)^{ - 1}}\, + \,c$) is correct because it is the result of integrating the given expression using substitution.