$\int t^{3 / 2} + t^{1 / 2} d t$

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Solution

$\int t^{3 / 2} + t^{1 / 2} d t \int t^{3 / 2} d t + \int t^{1 / 2} d t \frac{2}{5} t^{5 / 2} + \frac{2}{3} t^{3 / 2} + c$

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Let $θ \in (0, \frac{π}{4}) a n d t_{1} = (t a n θ)^{t a n θ}, t_{2} (t a n θ)^{c o t} t_{3} = (c o t θ)^{t a n θ} a n d t_{4} (c o t θ)^{t a n θ}$ , then

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∫t3/2+t1/2dt

∫t3/2+t1/2dt∫t3/2dt+∫t1/2dt25t5/2+23t3/2+c

$\int t^{3 / 2} + t^{1 / 2} d t$

$\int t^{3 / 2} + t^{1 / 2} d t \int t^{3 / 2} d t + \int t^{1 / 2} d t \frac{2}{5} t^{5 / 2} + \frac{2}{3} t^{3 / 2} + c$