"Evaluate the indefinite integral. ∫a+bx23ax+bx3dx" solved and explained

skomminbv

Answered question

2021-12-06

Evaluate the indefinite integral.

\int \frac{a + b x^{2}}{\sqrt{3 a x + b x^{3}}} d x

barcelodurazo0q

Beginner2021-12-07Added 13 answers

Step 1
Given integral:

\int \frac{a + b x^{2}}{\sqrt{3 a x + b x^{3}}} d x

Step 2
Now,
Apply u-substitution:

u = 3 a x + b x^{3}

Differentiate both sides:

d u = 3 a d x + 3 b x^{2} d x

\frac{d u}{3} = (a + b x^{2}) d x

Therefore,

\int \frac{a + b x^{2}}{\sqrt{3 a x + b x^{3}}} d x = \int \frac{d u}{3 \sqrt{u}}

= \frac{1}{3} \int u^{\frac{- 1}{2}} d u

Apply the Power Rule:

\int x^{a} d x = \frac{x^{a + 1}}{a + 1}, a \neq - 1

= \frac{1}{3} \cdot \frac{u^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1}

= \frac{2}{3} \sqrt{u}

Substitute back

u = 3 a x + b x^{3} :

= \frac{2}{3} \sqrt{3 a x + b x^{3}}

Add a constant to the solution:

= \frac{2}{3} \sqrt{3 a x + b x^{3}} + C

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