Indefinite integration practice · Medium

Option D is correct because the integral can be simplified by dividing by x³ and using a substitution, leading to a logarithmic and arctangent form.

The correct answer is $\frac{4}{3}$ because the integral of the odd part is zero, and the integral of the even part is calculated correctly.

Option C ($\frac{1}{e} \log \left(x^e+e^x\right)+c$) is correct because the derivative of the denominator is a constant multiple of the numerator.

Option D (($\tan x)^{8/3}$) is correct because by rewriting the integral in terms of $\tan x$ and then substituting, we can find the expression for H(x).

Option A ($x-\sqrt{1-x^2} \sin ^{-1} x+c$) is correct because substituting $\sin^{-1}x = t$ simplifies the integral, and integration by parts yields the correct result.

Option B ($\frac{1}{4}\,\,\sin \,2x\, - \,\frac{x}{2}\, + \,c$) is correct because it is the result of integrating sin(x) d(cos(x)) using substitution and trigonometric identities.

Option D (None of these) is incorrect. Option C ($\frac{1}{2} \log \left|\frac{\sqrt[27]{{\sec 27 \theta}}}{{\sec \theta}}\right|+C$) is the correct answer, obtained by using trigonometric identities to simplify the integral.

Option B is correct. The assertion and reason are both correct, but the reason doesn't directly explain the assertion. The derivative relates to the tangent's slope, and the indefinite integral represents parallel curves, but one doesn't explain the other.

Option A ($\frac{4}{3}\left( {\frac{{x - 1}}{{x + 2}}} \right)^{1/4} + c$) is correct because it is the result of evaluating the indefinite integral using substitution.

Option D ($\ln \sqrt{{p}+\sqrt{{p}^2-1}}-\frac{1}{2} \sec ^{-1} {p}$) is correct because it represents the correct integration of the given expression.

Option B ($x(\ln x)^n$) is correct because it represents the result of integration by parts applied to the given integral.

Option C ($\frac{{{{\cos }^5}\,\,x}}{5}\,\, - \,\,\frac{{{{\cos }^3}\,\,x}}{3}\,\, + \,\,c$) is correct because it results from correctly applying u-substitution with $u = \cos x$ to the given integral.

Option B ($\frac{1}{32}\{{-\frac{3}{2} \cos 2 x+\frac{1}{6} \cos 6 x\}}+C$) is correct because it is the result of integrating the given trigonometric expression after simplification.

Option A is correct because the integral is solved using integration by parts and the initial condition is applied to find the constant of integration, leading to the value of f(1).

Option B ($\frac{2}{3}{\left( {1 + x} \right)^{3/2}} + C$) is correct because the integral simplifies to $\int \sqrt{1+x} dx$, which integrates to the given expression.

Option D (${\sin ^{ - 1}}\,(\cos \,\mathrm{\theta })\, + \,c$) is correct because the integral can be directly computed using the definition of arcsin.

Option C ($\frac{{-1}}{{4}} \log _e|\frac{{(1+x^4)^{1 / 4}-x}}{{(1+x^4)^{1 / 4}+x}}|-\frac{{1}}{{2}} \operatorname{{Tan}}^{-1}(\frac{{(1+x^4)^{1 / 4}}}{{x}})+c$) is correct because after applying the appropriate substitution and simplifying, the integral evaluates to this form.

Option B ($\tan ^{-1} \sqrt{4 x+5}+c$) is correct because it is the result of integrating the given expression using substitution.

Option A ($\sinh ^{-1}\left(\frac{x^3+x^2-1}{2}\right)+c$) is correct because it results from completing the square and using the inverse hyperbolic sine integral.

Option C ($e^x\left(\sqrt{{\frac{{1+x^{{10}}}}{{1-x^{{10}}}}}}\right)+c$) is correct because the integral can be expressed in the form $\int e^x (f(x) + f'(x)) dx = e^x f(x) + c$.

Option C is correct because after substitution and simplification, the integral becomes a standard logarithmic form, resulting in $a = \frac{1}{3}$.

Option B ($\frac{1}{3}x - \ln \left( {{e^x} + 1} \right) + \frac{2}{3}\ln \left( {2{e^x} + 3} \right) + c$) is correct because after using partial fractions and substituting $e^x = t$, the integral evaluates to this expression.

Option D ($g(x) f(x)+c$) is correct because integrating by parts simplifies the expression to this result, given that $\int g(x) dx = g(x)$.

Option B ($-x \cot \frac{x}{2}+c$) is correct because it results from integrating the given expression after using trigonometric identities and integration by parts.

Option D ($- \frac{1}{{\sqrt 2 }}\cot \left( {\frac{x}{2} + \frac{\pi }{8}} \right)\\ + \\,c$) is the correct result after simplifying the integral using trigonometric identities and substitution.

Option B ($\frac{1}{2}\left[x^2 F\left(x^2\right)-\int F\left(x^2\right) d\left(x^2\right)\right]$) is correct because it uses integration by parts with appropriate substitutions.

Option C ($\frac{{(\tan^{-1}(\sin x+1))^2}}{{2}}+c$) is correct because it is the result of the integration after a suitable substitution.

Option B (e+1) is correct because it is the value obtained after evaluating the integral, finding f(x), and substituting x = -1 into the exponential of f(x).

Option A is correct because the integral simplifies to $\log(\sec x + \tan x) - 2 \tan(\frac{x}{2})$ after using trigonometric identities and partial fractions.

Option B ($\frac{{2}}{{3}}(1+x^2)^{3/2}+C$) is correct because it is the result of integrating the given function with respect to $x^2$.