Evaluate the following integral: ∫(4√(x + 4))/(4√(x + 4) + 4√(9 - x))dx, x ∈ [0, 5]

Question

Evaluate the following integral:

\(\int\limits_{0}^{5}\cfrac{\sqrt[4]{\text x+4}}{\sqrt[4]{\text x+4}+\sqrt [4]{9-\text x}}d\text x \)

Answer 1

Let us assume

I = \(\int\limits_{0}^{5}\cfrac{\sqrt[4]{\text x+4}}{\sqrt[4]{\text x+4}+\sqrt [4]{9-\text x}}d\text x \)... equation 1

By property we know that

I = \(\int\limits_{0}^{5}\cfrac{\sqrt[4]{9-\text x}}{\sqrt[4]{9-\text x}+\sqrt [4]{\text x+4}}d\text x \)...equation 2

Adding equation 1 and 2

2I = \(\int\limits_{0}^{5}\cfrac{\sqrt[4]{\text x+4}}{\sqrt[4]{\text x+4}+\sqrt [4]{9-\text x}}d\text x \) + \(\int\limits_{0}^{5}\cfrac{\sqrt[4]{9-\text x}}{\sqrt[4]{9-\text x}+\sqrt [4]{\text x+4}}d\text x \)

We know