7–64. Integration review Evaluate the following integrals.
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Question

7–64. Integration review Evaluate the following integrals.

57. ∫ dx / (x¹⸍² + x³⸍²)

Accepted Answer

Welcome back, everyone. Evaluate the integral integral of 1 divided by 2 X to the power of 1/2 plus 2 X to the power of 3 halves the X. For this problem, let's rewrite the original integral, which is DX. Divided by 2 X to the power of 1/2 plus 2X to the power of three halves. What we can do is simply begin with factorization. So we get integral of the X divided by 2. We can also take out X, raise the power of 1/2, and then print Cs this leaves us with 1 + X. Now let's introduce substitution. Let's suppose that u equals x to the power of 1/2. Then the derivative d u divided by d X is the derivative of X is the power of 1/2. Which is equal to 1/2 we bring down the exponent X to the power of -12. Or basically 1 divided by 2 X to the power of 1/2. So let's notice that we have that term in the denominator, which is 2x the power of 1/2. And what we can show is that Do you Is equal to. D X divided by 2 X to the power of 1/2. In other words, we're multiplying both sides by DX considering our derivative. And this is exactly what we have in our integral, the X divided by 2 X to the power of 1/2. So we're going to replace this whole ratio with DU. And then 1 + X becomes 1 + U2, right? Because U equals X is the power of 1/2, meaning U2 equals X, we can square both sides. So the integral that we want to evaluate is the integral of the EU. Divided by 1 plus U squad. And this integral can be evaluated using the tables of basic integrals. This is inverse of tangent of U + C. And now we can substitute for you. So there'd be inverse of tangent of you which is X of the power of one half. Or simply square root of x. So our final answer would be inverse of tangent of square root of x, plus a constant of integration C because we have an indefinite integral. Thank you for watching.

7–64. Integration review Evaluate the following integrals.
57. ∫ dx / (x¹⸍² + x³⸍²)

Key Concepts

Integration Techniques

Rational Functions

Limits of Integration

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