Indefinite integration practice · Medium

Option D (None of these) is correct because the provided solution appears to have an error in the initial problem statement and substitution, leading to an incorrect result. The question asks for a definite integral, but the provided solution calculates an indefinite integral.

Option B ($\log _e(1+\sqrt{2})-\frac{\pi}{4}$) is correct because after simplifying the integral and applying the initial condition $f(0)=0$, we find $f(1) = \log_e(1+\sqrt{2}) - \frac{\pi}{4}$.

Option A is correct because the integral can be simplified and solved by recognizing the form $\int e^x[f(x) + f'(x)]dx = e^x f(x) + c$.

Option D ($\tan \,\left( {x\\;{e^x}} \right)$) is correct because by substituting $t = xe^x$, the integral simplifies and can be easily solved.

Option D ($x e^{x+x^{-1}}+C$) is correct because it results from integrating by parts and simplifying the expression.

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

Option B is correct because the integral can be solved using integration by parts after a suitable substitution.

Option C ($x + \log \,\left( {\frac{{x - 1}}{{x + 1}}} \right) + c$) is correct because after performing the integration, the result matches this expression.

Option B ($\mathrm{x}^4\{8(\log x)^2-4(\log \mathrm{x})+1\}+\mathrm{C}$) is correct because it is the result of integrating the given expression using integration by parts twice.

Option A ($\frac{1}{2}\left[\frac{\left(x+\sqrt{x^2+1}\right)^{n+1}}{n+1}+\frac{\left(x+\sqrt{x^2+1}\right)^{n-1}}{n-1}\right]+c$) is correct because by substituting $t=x+\sqrt{x^2+1}$ and simplifying, we can evaluate the integral.

Option C ($f(x)\mathrm{g'}\,(x)\, - \,f'\,(x)\,\mathrm{g}\,(x)$) is correct because the integral simplifies to this expression after applying integration by parts twice.

Option C ($\log _e(f(x))+c$) is correct because by simplifying the limit and integrating, we can find the expression for the integral.

Option B ($\frac{{\sin ^6 x}}{{6}}-\frac{{\sin ^8 x}}{{8}}+c$) is correct because it is the result of integrating the given expression using substitution and trigonometric identities.

Option B (15 ( log | 1 + 53 tan ( x/2 )\ | )$\frac{1}{5}\,\,\left( {\log \,\,\left| {1\,\, + \,\,\frac{5}{3}\,{{\tan }\,\,\left( {x/2} \right)\}} \right|} \right)$) is correct because it is the result of integrating the given expression using the tangent half-angle substitution and applying the initial condition.

Option A ($a=-\frac{\pi}{4}, b \in R$) is correct because it is obtained by manipulating the integral using trigonometric identities and comparing the result with the given form.

The correct answer is A ($\frac{x}{6(x^6+1)}$) because after applying integration by parts and simplifying the expression, the result matches option A.

Option A is correct because the integral is solved by expressing the numerator as a linear combination of the denominator and its derivative.

Option A ($-\frac{1}{75}\left(1+\frac{5}{\mathrm{x}^{10}}\right)^{3 \cdot 2}+\mathrm{c}$) is correct because it is the result of the integration after a suitable substitution.

Option D (None of these) is correct because the integral cannot be solved using a simple substitution method, and the given options do not represent the correct antiderivative.

Option B ($\frac{1}{2} \tan ^{-1}(2\tan x)+c$) is correct because it is the result of the integration after dividing by $\cos^2 x$ and using a suitable substitution.

Option B ($\sqrt {2\,\,(\tan \,\,x\,\, + \,\,\tan \,\,\mathrm{\theta })\,\,\sec \,\,\mathrm{\theta }} + c$) is correct because it is the result of correctly evaluating the integral after trigonometric simplification and substitution.

Option B ($\frac{\pi }{2}\\ - \\1$) is correct because the integral can be solved by rationalizing the denominator and then using trigonometric substitution or direct integration.

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

Option C is correct because the integral simplifies to $\log \,\,\left| {1 - \cot \,\,\frac{x}{2}} \right| + c$ after using trigonometric identities and substitution.

Option B ($\frac{x}{2}\, + \,d$) is correct because differentiating $\log \,{\left\{ {f\left( x \right)} \right\}^2}\, + \,c$ gives $\frac{1}{{f\left( x \right)}}$ when $f(x) = \frac{x}{2} + d$.

Option B ($\frac{\mathrm{\pi }}{6}$) is correct because after substitution, the integral becomes a standard arcsin integral, evaluating to $\frac{\pi}{6}$.

Option A is correct because the integral simplifies to a form that can be solved by substitution, resulting in the given expression.

Option B ($-12 \log _e|1-x|-3 x+c$) is correct because it is the result of integrating f(x) after finding the expression for f(x) by substituting $y = \frac{x-4}{x+2}$.

Option A (${\log _e}\sqrt[5]{{\frac{{\tan \\,x - 2}}{{2\tan \\,x + 1}}}}\\ + \\ c$) is correct because the integral is solved by dividing by cos²(x) and using partial fractions.

Option A is correct because the integral can be solved by multiplying the numerator and denominator by $e^x$ and then using partial fractions.