Indefinite integration practice · Mixed difficulty

Option B is correct because the integral simplifies to x / (a² * sqrt(a² - b² * x²)) + C after trigonometric substitution.

Option A ($\frac{1}{{ab}}\,{\tan ^{ - 1}}\,\,\left( {\frac{{a\,\,\tan \,\,x}}{b}} \right)\,\, + \,\,c$) is correct because the integral can be solved by dividing by cos²(x) and using a suitable substitution.

Option A ($x\;{e^{{{\tan }^{ - 1}}x}}\, + \,C$) is correct because it results from a u-substitution and simplification of the integral.

Option C is correct because the integral is solved using the tangent half-angle substitution, resulting in $\frac{2}{3} \tan^{-1}(\frac{1}{3} \tan \frac{x}{2})+c$.

Option B ($x e^{x+\frac{{1}}{{x}}}+C$) is correct because it represents the result of the given integral after applying integration by parts and simplification.

Option B is correct because the integral simplifies to a logarithmic function after simplification.

Option C ($\log _{e} \frac{(\pi+4)^{2}}{32}-\frac{\pi^{2}}{4(\pi+4)}$) is correct because it is the result of evaluating the definite integral with the given initial condition.

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.

Option C ($\log x-\log (1-x)+c$) is correct because the integral is solved by partial fraction decomposition.

Option A ($-\frac{1}{4} \cos ^4 x^2+c$) is correct because it is the result of integrating the given expression using substitution.

Option A ($\frac{1}{2}\left[\log \left(x+\sqrt{1+x^2}\right)\right]^2+c$) is correct because it is the result of the integration after using a suitable substitution.

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 A is correct because the integral simplifies to $-\frac{{1}}{{2(e^{{2x}}+1)}}+c$ after substitution and integration.

Option B ($\left( {2x\, - \,4\,\sqrt x + 4} \right){e^{\sqrt x }} + \,c$) is correct because it is the result of integrating the given expression using substitution and integration by parts.

Option B ($\frac{1}{2}\,{\mathrm{g}^2}\left( x \right)$) is correct because after applying integration by parts and simplifying, the result matches this option.

Option B ($\frac{{5}}{{2}} \sqrt{{1+x^{4 / 5}}}+C$) is correct because it is the result of the integration after appropriate substitution.

Option B ($\ - \frac{3}{8}\,{\left( {\frac{1}{{{x^2}}}\,\, - \,\,1} \right)^{4/3}} + \,\,C$) is correct because after simplifying the integrand and using substitution, the result matches this option.

The correct answer is D ($\frac{2}{{{a^2}}}\,\\,\,\,\left( {x\,\, + \,\,\frac{b}{a}\,\\,\,\log \,\,\,\left( {ax\,\, + \,\,b} \right)} \right)\,\, - \,\,\,\frac{{{{{x^2}}}}}{{{{a\left( {ax\,\, + \,\,b} \right)}}}}\,\, + \,\,c$) because it results from correct substitution and integration by parts.

Option C ($-\frac{1}{(\sec x+\tan x)^{11 / 2}}\{\frac{1}{11}+\frac{1}{7}(\sec x+\tan x)^2\}+K$) is correct because substituting $t = \sec x + \tan x$ and simplifying leads to the given result.

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 B is correct because substituting $t = 3x - 5$ transforms the integral to $\frac{1}{3} \int \tan t \sec t dt$, which equals $\frac{1}{3} \sec t + c = \frac{1}{3} \sec(3x - 5) + c$.

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, $\sqrt{2} \sin ^{-1}(\sin x-\cos x)+C$, is correct because manipulating the integrand and using a trigonometric identity leads to this 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 C is correct because the integral simplifies to \$\log \sec x + \log (\sec x + \tan x)$, which can be combined to give \$\log \sec x(\sec x + \tan x)$.

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 B, $-\sqrt{2} \tan ^{-1} \frac{x}{\sqrt{2}}+\sqrt{3} \tan ^{-1} \frac{x}{\sqrt{3}}+c$, is the correct answer because it results from partial fraction decomposition and integration of the given expression.

Option A (${e^x}f\,\left( x \right) + c$) is correct because the integral of ${e^x}[f(x) + f'(x)] dx$ results in ${e^x}f(x) + c$ using integration by parts.