Indefinite integration practice · Mixed difficulty

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 C (12) is correct because by using vector triple product and given conditions, we can simplify the expression and find its magnitude.

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

Option D ($\frac{{5(2 \sin \theta + 1)}}{{(\sin \theta + 3)}}$) is correct because it matches the simplified form of the integral after substitution and partial fraction decomposition.

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 B ($x{e}^{{x}+\frac{1}{x}}+{c}$) is correct because the integral can be solved using integration by parts after rewriting the expression.

Option B is correct because the integral is solved using substitution, resulting in the given expression with the correct constant of integration.

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 is correct because the integral of the given function simplifies to the logarithm of the absolute value of (1 + cos x) / cos x.

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

The correct answer is f(x) = log (x + √(x² + 1)) and g(x) = x²/2 because after substitution and integration, the given integral evaluates to this result.

Option C ($\log \tan \theta+\frac{1}{2} \tan ^2 \theta+c$) is correct because the integral simplifies to this form after substitution and integration.

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 ($\frac{{1}}{{5}}{{\operatorname{{Tan}} ^{ - 1}}}\left( {{{{x^5} + \frac{{1}}{{{{x^5}}}}}}} \right) + c$) is correct because it is the result of the integration after 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 B ($\frac{{1}}{{16}}\left[ {4{x^4}\log x - {x^4}} \right] + c$) is correct because it is the result of integrating the given expression using integration by parts.

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 A (x + c) is correct because the integral simplifies to arcsin(sin(x)) + c, which equals x + c.

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.

Option D ($\tan^{-1}(-2 \cot 2x) + C$) is correct because the integral simplifies to this form after applying trigonometric identities and substitution.

Option B ($\frac{{1}}{{3}}(\log x)^3+c$) is correct because the integral simplifies to this form after using properties of logarithms and substitution.

Option C ($y=\cot ^{-1} x$) is correct because it is derived by taking the limit as h approaches 0 and solving for y.