Common integrals Quiz 1

Question 1

If $$I\,=\,\int{\left( 3{{x}^{2}}+125x+64 \right)dx}$$, then what is $$\int_{0}^{1}{I}\,dx$$ equal to?

Accepted Answer

$$\dfrac{225}{2}$$

Answer

$$\dfrac{159}{2}$$

Answer

$$\dfrac{171}{2}$$

Answer

$$\dfrac{221}{2}$$

Question 2

Write the value of the integral $$\int{{{\left( \sqrt{3x+1}+\dfrac{1}{\sqrt{3x+1}} \right)}^{2}}\,dx}$$.

Accepted Answer

$$\dfrac{3{{x}^{2}}}{2}\,+\,x+\dfrac{1}{3}\,\,\log \,\,\left| 3x\,+\,1 \right|\,+2x+c$$

Answer

$$\dfrac{3{{x}^{2}}}{2}+\dfrac{1}{3}\log \,\left| 3x-1 \right|+x-2x+c$$

Answer

$$\dfrac{3{{x}^{2}}}{2}\,-\,\dfrac{1}{3}\,\,\log \,\,\left| 3x\,-\,1 \right|\,+2x+c$$

Answer

$$\dfrac{3{{x}^{2}}}{2}\,+\,\dfrac{1}{3}\,\,\log \,\,\left| 3x\,+\,1 \right|\,+2x+c$$

Question 3

Evaluate the integral $$\int{{{e}^{\log \sqrt{29-9x}}}\,dx}$$.

Accepted Answer

$$\dfrac{-2}{27}\,{\left( 29-9x \right)}^{\frac{3}{2}}+c$$

Answer

$$\dfrac{2}{27}\,{\left( 29+9x \right)}^{\frac{3}{2}}+c$$

Answer

$$\dfrac{2}{27}\,{\left( -29+2x \right)}^{\frac{3}{2}}+c$$

Answer

$$\dfrac{-2}{27}\,{\left( -29-2x \right)}^{\frac{3}{2}}+c$$

Question 4

Integrate the function $$\sqrt{\dfrac{25-25\cos 9x}{25+25\cos 9x}}$$ with respect to.

Accepted Answer

$$\dfrac{2}{9}\,\log \,\,\left| \sec \,\dfrac{9x}{2} \right|+c$$

Answer

$$\dfrac{9}{2}\,\log \,\,\left| \cot \dfrac{9x}{2} \right|+c$$

Answer

$$\dfrac{2}{9}\,\log \,\,\left| \sin \,\dfrac{9x}{2} \right|+c$$

Answer

$$\dfrac{9}{2}\,\log \,\,\left| \sec \,\dfrac{9x}{2} \right|+c$$

Question 5

Write the value of integral $$z=\int{\dfrac{{{\sin }^{2}}5x+{{\cos }^{2}}5x}{{{\sin }^{-1}}5x+{{\cos }^{-1}}5x}dx}$$.

Accepted Answer

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

Answer

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

Answer

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

Answer

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