Introduction to integrals Quiz 1

Question 1

Integrate the following term: $$\int {\dfrac{1}{{\sqrt {8x + 5}  + \sqrt {8x + 4} }}dx} $$.

Accepted Answer

$$\dfrac{1}{{12}}\left[ {{{\left( {8x + 5} \right)}^{\dfrac{3}{2}}} - {{\left( {8x + 4} \right)}^{\dfrac{3}{2}}}} \right] + C$$

Answer

$$\dfrac{1}{6}\left[ {{{\left( {8x + 5} \right)}^{\dfrac{3}{2}}} - {{\left( {8x + 4} \right)}^{\dfrac{3}{2}}}} \right] + C$$

Answer

$$\dfrac{1}{2}\left[ {{{\left( {5x + 8} \right)}^{\dfrac{3}{2}}} - {{\left( {8x + 4} \right)}^{\dfrac{3}{2}}}} \right] + C$$

Answer

$$\dfrac{1}{{12}}\left[ {{{\left( {5x + 8} \right)}^{\dfrac{3}{2}}} - {{\left( {4x + 8} \right)}^{\dfrac{3}{2}}}} \right] + C$$

Question 2

Integrate the following term: $$\int {\dfrac{{2x + 1}}{{x + {x^2} + 4}}dx} $$.

Accepted Answer

$$\log \left| {{x^2} + x + 4} \right| + C$$

Answer

$$\dfrac{1}{2}\log \left| {{x^2} + x + 4} \right| + C$$

Answer

$$\dfrac{1}{8}\log \left| {{x^2} + x + 4} \right| + C$$

Answer

$$\dfrac{1}{4}\log \left| {{x^2} + x + 4} \right| + C$$

Question 3

Find the integration of $$\int {\dfrac{{{{\left( {{x^6} - x} \right)}^{\dfrac{1}{6}}}}}{{{x^7}}}dx} $$

Accepted Answer

$$\dfrac{6}{{35}}{\left( {1 - {x^{ - 5}}} \right)^{\dfrac{7}{6}}} + C$$

Answer

$$\dfrac{3}{{14}}{\left( {1 - {x^{ - 5}}} \right)^{\dfrac{7}{6}}} + C$$

Answer

$$\dfrac{3}{{14}}{\left( {1 - {x^{ - 5}}} \right)^{\dfrac{6}{7}}} + C$$

Answer

$$\dfrac{3}{{14}}{\left( {1 - {x^{ - 4}}} \right)^{\dfrac{7}{6}}} + C$$

Question 4

Find the integral of  $$\int {\dfrac{{\sin \sqrt x }}{{\sqrt x }}dx} $$.

Accepted Answer

$$ - 2\cos \sqrt x  + C$$

Answer

$$\dfrac{1}{2}\cos \sqrt x  + C$$

Answer

$$2\cos \sqrt x  + C$$

Answer

None

Question 5

Find the integration of $$\int {\cos ecx\cot x + 	an x\sec xdx} $$

Accepted Answer

$$ - \dfrac{1}{{\sin x}} + \dfrac{1}{{\cos x}} + C$$

Answer

$$ - \dfrac{1}{{\sec x}} + \dfrac{1}{{\cos ecx}} + C$$

Answer

$$\dfrac{1}{{\sin x}} - \dfrac{1}{{\cos x}} + C$$

Answer

$$\dfrac{1}{{\sec x}} - \dfrac{1}{{\cos ecx}} + C$$