Integration by substitution, by parts and by partial fractions — practice questions

Question 1 · difficulty L2 · understanding

The integral $$\int {{{(2x - 1)\cos \sqrt {{{(2x - 1)}^2} + 5} } \over {\sqrt {4{x^2} - 4x + 6} }}} dx$$ is equal to (where c is a constant of integration)

• A. $${1 \over 2}\sin \sqrt {{{(2x - 1)}^2} + 5} + c$$
• B. $${1 \over 2}\cos \sqrt {{{(2x + 1)}^2} + 5} + c$$
• C. $${1 \over 2}\cos \sqrt {{{(2x - 1)}^2} + 5} + c$$
• D. $${1 \over 2}\sin \sqrt {{{(2x + 1)}^2} + 5} + c$$

Question 2 · difficulty L3 · understanding

If $\int(\sin x)^{\frac{-11}{2}}(\cos x)^{\frac{-5}{2}} d x= -\frac{p_1}{q_1}(\cot x)^{\frac{9}{2}}-\frac{p_2}{q_2}(\cot x)^{\frac{5}{2}}-\frac{p_3}{q_3}(\cot x)^{\frac{1}{2}}+\frac{p_4}{q_4}(\cot x)^{\frac{-3}{2}}+\mathrm{C}$, where $p_i$ and $q_i$ are positive integers with $\operatorname{gcd}\left(p_i, q_i\right)=1$ for $i=1,2,3,4$ and C is the constant of integration, then $\frac{15 p_1 p_2 p_3 p_4}{q_1 q_2 q_3 q_4}$ is equal to $\_\_\_\_$

Question 3 · difficulty L3 · understanding

If $\int\left(\frac{1}{x}+\frac{1}{x^3}\right)\left(\sqrt[23]{3 x^{-24}+x^{-26}}\right) \mathrm{d} x=-\frac{\alpha}{3(\alpha+1)}\left(3 x^\beta+x^\gamma\right)^{\frac{\alpha+1}{\alpha}}+C, x>0,(\alpha, \beta, \gamma \in \mathbf{Z})$, where C is the constant of integration, then $\alpha+\beta+\gamma$ is equal to ___________.

Question 4 · difficulty L3 · understanding

If $$\int \frac{1}{\sqrt[5]{(x-1)^4(x+3)^6}} \mathrm{~d} x=\mathrm{A}\left(\frac{\alpha x-1}{\beta x+3}\right)^B+\mathrm{C}$$, where $$\mathrm{C}$$ is the constant of integration, then the value of $$\alpha+\beta+20 \mathrm{AB}$$ is _________.

Question 5 · difficulty L3 · understanding

If $$\int \operatorname{cosec}^5 x d x=\alpha \cot x \operatorname{cosec} x\left(\operatorname{cosec}^2 x+\frac{3}{2}\right)+\beta \log _x\left|\tan \frac{x}{2}\right|+\mathrm{C}$$ where $$\alpha, \beta \in \mathbb{R}$$ and $$\mathrm{C}$$ is the constant of integration, then the value of $$8(\alpha+\beta)$$ equals _________.

Question 6 · difficulty L3 · understanding

Let $$I(x)=\int \sqrt{\frac{x+7}{x}} \mathrm{~d} x$$ and $$I(9)=12+7 \log _{e} 7$$. If $$I(1)=\alpha+7 \log _{e}(1+2 \sqrt{2})$$, then $$\alpha^{4}$$ is equal to _________.

Question 7 · difficulty L3 · understanding

If $$\int {{{\sin x} \over {{{\sin }^3}x + {{\cos }^3}x}}dx = }$$ $$\alpha {\log _e}|1 + \tan x| + \beta {\log _e}|1 - \tan x + {\tan ^2}x| + \gamma {\tan ^{ - 1}}\left( {{{2\tan x - 1} \over {\sqrt 3 }}} \right) + C$$, when C is constant of integration, then the value of $$18(\alpha + \beta + {\gamma ^2})$$ is ______________.

Question 8 · difficulty L3 · understanding

If $$\int {{{2{e^x} + 3{e^{ - x}}} \over {4{e^x} + 7{e^{ - x}}}}dx = {1 \over {14}}(ux + v{{\log }_e}(4{e^x} + 7{e^{ - x}})) + C}$$, where C is a constant of integration, then u + v is equal to _____________.

Question 9 · difficulty L3 · understanding

If $$\int {{{dx} \over {{{({x^2} + x + 1)}^2}}} = a{{\tan }^{ - 1}}\left( {{{2x + 1} \over {\sqrt 3 }}} \right) + b\left( {{{2x + 1} \over {{x^2} + x + 1}}} \right) + C}$$, x > 0 where C is the constant of integration, then the value of $$9\left( {\sqrt 3 a + b} \right)$$ is equal to _____________.

Question 10 · difficulty L3 · understanding

If $$f(x) = \int {{{5{x^8} + 7{x^6}} \over {{{({x^2} + 1 + 2{x^7})}^2}}}dx,(x \ge 0),f(0) = 0}$$ and $$f(1) = {1 \over K}$$, then the value of K is