∫xln⁡$$aax/23a5x/2b3x+ln$$⁡$$bbx2a2xb4xdx$$ where a,b∈$$R+is$$ equal to

Correct option is 4−16ln⁡a2b31a2xb3xln⁡a2xb3xe+k

Questions

$\int x (\frac{\ln a^{a^{x / 2}}}{3 a^{5 x / 2} b^{3 x}} + \frac{\ln b^{b^{x}}}{2 a^{2 x} b^{4 x}}) d x (where a, b \in R^{+}) is equal to$

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16ln⁡a2b3a2xb3xln⁡a2xb3xe+k

16ln⁡a2b31a2xb3xln⁡1ea2xb3x+k

16ln⁡a2b31a2xb3xln⁡a2xb3xe+k

−16ln⁡a2b31a2xb3xln⁡a2xb3xe+k

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detailed solution

Correct option is D

I=∫xln⁡aax/23a5x/2b3x+ln⁡bbx2a2xb4xdx=∫ln⁡a2xb3x6a2xb3xdx Let a2xb3x=t. Then tln⁡a2b3dx=dt. Therefore, I=∫16ln⁡a2b3ln⁡tt2dt=16ln⁡a2b3−ln⁡tt−∫−1t2dt=−16ln⁡a2b3ln⁡ett+k=−16ln⁡a2b3ln⁡a2xb3xea2xb3x+k

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∫xln⁡aax/23a5x/2b3x+ln⁡bbx2a2xb4xdx where a,b∈R+is equal to

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$\int x (\frac{\ln a^{a^{x / 2}}}{3 a^{5 x / 2} b^{3 x}} + \frac{\ln b^{b^{x}}}{2 a^{2 x} b^{4 x}}) d x (where a, b \in R^{+}) is equal to$