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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

166.
$\int (\sqrt[3]{x}) (\sqrt[5]{1 + \sqrt[3]{x^{4}}}) dx$ is equal to

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

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

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

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

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

$Let I = \int (\sqrt[3]{x}) (\sqrt[5]{1 + \sqrt[3]{x^{4}}}) dx Put \sqrt[3]{x^{4}} = t \Rightarrow \frac{4}{3} \sqrt[3]{x} dx = dt ∴ I = \frac{3}{4} (\sqrt[5]{1 + t}) dt = \frac{3}{4} [\frac{{(1 + t)}^{\frac{1}{5} + 1}}{\frac{1}{5} + 1}] + c = \frac{5}{8} [{(1 + \sqrt[3]{x^{4}})}^{\frac{6}{5}}] + c$

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