Integration by parts Quiz 1

Question 1

Simplify the integral $$\int{{{e}^{\log \,{{x}^{5}}}}}.\,\,\log \,x\,\,dx$$

Accepted Answer

$$\dfrac{{{x}^{6}}\,\log \,x}{6}-\dfrac{{{x}^{6}}}{36}+c$$

Answer

$$\dfrac{{{x}^{5}}\log \,x}{5}+\dfrac{{{x}^{6}}}{36}+c$$

Answer

$$\dfrac{{{x}^{6}}\,\log \,x}{6}\,+\,\dfrac{{{x}^{6}}}{36}+c$$

Answer

$$\dfrac{{{x}^{5}}\,\log \,x}{5}-\dfrac{{{x}^{6}}}{36}+c$$

Question 2

Evaluate $$2m$$ where $$m=\,\int{{{t}^{25}}\log \,19t\,\,dt}$$.

Accepted Answer

$$\dfrac{{{t}^{26}}\log \,19t}{13}-\dfrac{{{t}^{26}}}{338}+\dfrac{c}{2}$$

Answer

$$\dfrac{{{t}^{26}}\,\log \,19t}{13}+\dfrac{{{t}^{26}}}{338}+\dfrac{c}{2}$$

Answer

$$\dfrac{{{t}^{26}}\,\log \,19t}{32}+\dfrac{{{t}^{52}}}{538}+\dfrac{c}{8}$$

Answer

$$\dfrac{{{t}^{26}}\,\log \,19t}{26}-\dfrac{{{t}^{52}}}{538}+\dfrac{c}{4}$$

Question 3

Evaluate $$\int{\left( 2m+9 \right)\,\,sin\,\left( 2m\,+9 \right)}\,dm$$.

Accepted Answer

$$\dfrac{-\left( 2m+9 \right)\,\cos \,\left( 2m+9 \right)}{2}+\dfrac{\sin \,\left( 2m+9 \right)}{2}+c$$

Answer

$$\dfrac{-\left( 2m+9 \right)\,\cos \,\left( 2m\,+\,9 \right)}{2}-\dfrac{sin\left( 2m+9 \right)}{2}+c$$

Answer

$$\dfrac{-\left( 2m+9 \right)\,\cos \,\left( 2m+9 \right)}{2}+\dfrac{\sin \,\left( 2m-9 \right)}{2}+c$$

Answer

$$\dfrac{\left( 2m+9 \right)\,\cos \,\left( 2m+9 \right)}{2}-\dfrac{\sin \,\left( 2m-9 \right)}{2}+c$$

Question 4

What is the value of the integral$$\int{x.\left[ {{\sec }^{-1}}\left( 7{{x}^{2}}+5x-1 \right)+\,{{\sin }^{-1}}\left( \dfrac{1}{7{{x}^{2}}+5x-1} \right) \right]}\,dx?$$

Accepted Answer

$$\dfrac{\pi }{4}{{x}^{2}}+c$$

Answer

$$\dfrac{\pi }{4}x+c$$

Answer

$$\dfrac{\pi }{6}x\,+c$$

Answer

$$\dfrac{\pi }{12}x\,+c$$

Question 5

Integrate:$$\int{\dfrac{{{	an }^{-1}}\left( 11h+12 ight).\,{{e}^{{{	an }^{-1}}\left( 11h+12 ight)}}}{1+{{\left( 11\,h+12 ight)}^{2}}}dh}$$.

Accepted Answer

$$\dfrac{1}{11}\left[ {{	an }^{-1}}\left( 11h+12 ight).\,{{e}^{{{	an }^{-1}}\left( 11h+12 ight)}}-{{e}^{{{	an }^{-1}}\left( 11h+12 ight)}}+c ight]$$

Answer

$$\dfrac{1}{11}\left[ {{	an }^{-1}}\left( 11h+12 ight).\,{{e}^{{{	an }^{-1}}\left( 11h+12 ight)}}+c ight]$$

Answer

$$\dfrac{1}{11}\left[ {{e}^{ta{{n}^{-1}}\left( 11h+12 ight)}}+\,{{	an }^{-1}}\left( 11h+12 ight)+c ight]$$

Answer

$$\dfrac{1}{11}\left[ {{	an }^{-1}}\left( 11h+12 ight).{{e}^{ta{{n}^{-1}}\left( 11h+12 ight)}}+2{{	an }^{-1}}h\,+c ight]$$