Integral as an antiderivative; fundamental integrals — practice questions

Question 1 · difficulty L3 · understanding

Let $\mathrm{I}(x)=\int \frac{d x}{(x-11)^{\frac{11}{13}}(x+15)^{\frac{15}{13}}}$. If $\mathrm{I}(37)-\mathrm{I}(24)=\frac{1}{4}\left(\frac{1}{\mathrm{~b}^{\frac{1}{13}}}-\frac{1}{\mathrm{c}^{\frac{1}{13}}}\right), \mathrm{b}, \mathrm{c} \in \mathcal{N}$, then $3(\mathrm{~b}+\mathrm{c})$ is equal to

• A. 39
• B. 22
• C. 40
• D. 26

Question 2 · difficulty L3 · understanding

If $$\int \frac{1}{\mathrm{a}^2 \sin ^2 x+\mathrm{b}^2 \cos ^2 x} \mathrm{~d} x=\frac{1}{12} \tan ^{-1}(3 \tan x)+$$ constant, then the maximum value of $$\mathrm{a} \sin x+\mathrm{b} \cos x$$, is :

• A. $$\sqrt{41}$$
• B. $$\sqrt{39}$$
• C. $$\sqrt{40}$$
• D. $$\sqrt{42}$$

Question 3 · difficulty L3 · understanding

For $$\alpha, \beta, \gamma, \delta \in \mathbb{N}$$, if $$\int\left(\left(\frac{x}{e}\right)^{2 x}+\left(\frac{e}{x}\right)^{2 x}\right) \log _{e} x d x=\frac{1}{\alpha}\left(\frac{x}{e}\right)^{\beta x}-\frac{1}{\gamma}\left(\frac{e}{x}\right)^{\delta x}+C$$ , where e=\sum_\limits{n=0}^{\infty} \frac{1}{n !} and $$\mathrm{C}$$ is constant of integration, then $$\alpha+2 \beta+3 \gamma-4 \delta$$ is equal to :

• A. $$-8$$
• B. $$-4$$
• C. 1
• D. 4

Question 4 · difficulty L3 · understanding

If $$\int {{{({x^2} + 1){e^x}} \over {{{(x + 1)}^2}}}dx = f(x){e^x} + C}$$, where C is a constant, then $${{{d^3}f} \over {d{x^3}}}$$ at x = 1 is equal to :

• A. $$- {3 \over 4}$$
• B. $${3 \over 4}$$
• C. $$- {3 \over 2}$$
• D. $${3 \over 2}$$

Question 5 · difficulty L4 · understanding

The integral $$\int\left[\left(\frac{x}{2}\right)^x+\left(\frac{2}{x}\right)^x\right] \ln \left(\frac{e x}{2}\right) d x$$ is equal to :

• A. $$\left(\frac{x}{2}\right)^{x}+\left(\frac{2}{x}\right)^{x}+C$$
• B. $$\left(\frac{x}{2}\right)^{x}-\left(\frac{2}{x}\right)^{x}+C$$
• C. $$\left(\frac{x}{2}\right)^{x} \log _{2}\left(\frac{2}{x}\right)+C$$
• D. None

Question 6 · difficulty L4 · understanding

$$\text { The integral } \int \frac{\left(1-\frac{1}{\sqrt{3}}\right)(\cos x-\sin x)}{\left(1+\frac{2}{\sqrt{3}} \sin 2 x\right)} d x \text { is equal to }$$

• A. $$\frac{1}{2} \log _{e}\left|\frac{\tan \left(\frac{x}{2}+\frac{\pi}{12}\right)}{\tan \left(\frac{x}{2}+\frac{\pi}{6}\right)}\right|+C$$
• B. $$\frac{1}{2} \log _{e}\left|\frac{\tan \left(\frac{x}{2}+\frac{\pi}{6}\right)}{\tan \left(\frac{x}{2}+\frac{\pi}{3}\right)}\right|+C$$
• C. $$\log _{e}\left|\frac{\tan \left(\frac{x}{2}+\frac{\pi}{6}\right)}{\tan \left(\frac{x}{2}+\frac{\pi}{12}\right)}\right|+C$$
• D. $$\frac{1}{2} \log _{e}\left|\frac{\tan \left(\frac{x}{2}-\frac{\pi}{12}\right)}{\tan \left(\frac{x}{2}-\frac{\pi}{6}\right)}\right|+C$$