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Integrals

Multiple Choice Questions

161.

$\int e^{3 \log (x)} {(x^{4} + 1)}^{- 1} dx$ is equal to

e³^log(x) + c
$\frac{1}{4} \log (x^{4} + 1)$
log(x⁴ + 1) + c
$\frac{1}{2} \log (x^{4} + 1) + c$

162.

If $\int_{0}^{π} xf (\sin (x)) dx = A \int_{0}^{\frac{π}{2}} f (\sin (x)) dx$ , then A is

0
$π$
$\frac{π}{4}$
2 $π$

163.

$\int_{- 2}^{2} |[x]| dx$ is equal to

164.

If $\int \frac{\sin (x)}{\sin (x - α)} dx = Ax + Blog (\sin (x - α)) + C$ , then value of A - B at $α = \frac{π}{2}$ is

165.
If $\int_{a}^{b} x^{3} dx = 0 and \int_{a}^{b} x^{2} dx = \frac{2}{3}$ , then the values of a and b are respectively

1, - 1

- 1, 1

1, 1

- 1, - 1

- 1, 1

$Since, \int_{a}^{b} x^{3} dx = 0 and \int_{a}^{b} x^{2} dx = \frac{2}{3} ∴ {[\frac{x^{4}}{4}]}_{a}^{b} = 0 and {[\frac{x^{3}}{3}]}_{a}^{b} = \frac{2}{3} \Rightarrow b^{4} - a^{4} = 0 and b^{3} - a^{3} = 2 \Rightarrow a = - 1, b = 1$

166.

$\int (\sqrt[3]{x}) (\sqrt[5]{1 + \sqrt[3]{x^{4}}}) dx$ is equal to

${(1 + x^{\frac{3}{4}})}^{\frac{6}{5}} + c$
${(1 + x^{\frac{4}{3}})}^{\frac{6}{5}} + c$
$\frac{5}{8} {(1 + x^{\frac{4}{3}})}^{\frac{6}{5}} + c$
$\frac{1}{6} {(1 + x^{\frac{4}{3}})}^{6} + c$

167.

If u = - $f'' (θ) \sin (θ) + f' (θ) \cos (θ)$ and v = $f'' (θ) \cos (θ) + f' (θ) \sin (θ)$ , then $\int {[{(\frac{du}{dθ})}^{2} + {(\frac{dv}{dθ})}^{2}]}^{\frac{1}{2}} dθ$ is equal to