Logarithmic integration Quiz 1

Question 1

Integrate the function $$y=\left( 3{{x}^{2}}-7x \right)\log \left( 2{{x}^{3}}-7{{x}^{2}}-9 \right)$$ with respect to $$x$$.

Accepted Answer

$$\dfrac{1}{2}\left( 2{{x}^{3}}-7{{x}^{2}}-9 \right)\left[ \log \left( 2{{x}^{3}}-7{{x}^{2}}-9 \right)-1 \right]+c$$

Answer

$$\left( 6{{x}^{3}}-7{{x}^{2}}-9 \right)\left[ \log \left( 2{{x}^{3}}-7{{x}^{2}}-9 \right)-1 \right]+c$$

Answer

$$\dfrac{3x}{2{{x}^{3}}-7{{x}^{2}}-9}+c$$

Answer

$$\dfrac{1}{2}\left( 6{{x}^{3}}-7{{x}^{2}}-9 \right)\left[ \log \left( 6{{x}^{3}}-7{{x}^{2}}-9 \right)-1 \right]+c$$

Question 2

Find the antiderivative of $$xy={{\left( \log x \right)}^{-1}}$$ with respect to $$x$$.

Accepted Answer

$$\log \left( \log x \right)+c$$

Answer

$$\log x\left( \log x \right)+c$$

Answer

$$\log {{\left( \log x \right)}^{2}}+c$$

Answer

$${{\left( \log x \right)}^{2}}+c$$

Question 3

Evaluate: $$I=\int{{{t}^{2}}\log \left( {{t}^{3}}+b \right)dt}$$ .

Accepted Answer

$$\dfrac{\left( {{t}^{3}}+b \right)}{3}\left[ \log \left( {{t}^{3}}+b \right)-1 \right]+c$$

Answer

$$\dfrac{t}{3}\left[ \log \left( {{t}^{3}}+b \right)-1 \right]+c$$

Answer

$$\dfrac{{{t}^{3}}}{3}\left[ \log \left( {{t}^{3}}+b \right)-1 \right]+c$$

Answer

$$\dfrac{3{{t}^{2}}}{3}\left[ \log \left( {{t}^{3}}+b \right)-1 \right]+c$$

Question 4

Evaluate: $$\int{\sqrt{\cos x\cot x}\log \left( \sqrt{\sin x} \right)dx}$$.

Accepted Answer

$$2\sqrt{\sin x}\left[ \log \left( \sqrt{\sin x} \right)-1 \right]+c$$

Answer

$$2\sqrt{\log x}\left[ \log \left( \sqrt{\sin x} \right)-1 \right]+c$$

Answer

$$2\sqrt{\sin x}\left[ \left( \sqrt{\sin (\log x)} \right)-1 \right]+c$$

Answer

$$2\log (\sin x)\left[ \log \left( \sqrt{\sin x} \right)-1 \right]+c$$

Question 5

Integrate the function $$\dfrac{y+7x}{{{x}^{2}}}=\left( \log x \right)$$ with respect to $$x$$ and determine which of the following option is true?

Accepted Answer

$$\int{ydx}=\dfrac{1}{3}x^3\ln \left(x\right)-\dfrac{x^3}{9}-\dfrac{7x^2}{2}+C$$

Answer

$$\int{ydx}=\dfrac{1}{3}x^3\ln \left(x\right)+\dfrac{x^3}{9}+\dfrac{7x^2}{2}+C$$

Answer

$$\int{ydx}=\dfrac{{{x}^{2}}}{9}\left( x\log x-\dfrac{1}{3} \right)+\dfrac{7{{x}^{2}}}{2}+c$$

Answer

$$\int{ydx}=\dfrac{{{x}^{2}}}{2}\left( \log x-\dfrac{1}{3}x \right)-\dfrac{7{{x}^{3}}}{3}+c$$