Evaluate the following integral: ∫ (3√x)/(3√x + 3√(7 - x))dx, x ∈ [0, 7]

Question

Evaluate the following integral: 

\(\int\limits_{0}^{7}\cfrac{\sqrt[3]{\text x}}{\sqrt[3]{\text x}+\sqrt[3]{7-\text x}}d\text x \)

Answer 1

Let us assume

I = \(\int\limits_{0}^{7}\cfrac{\sqrt[3]{\text x}}{\sqrt[3]{\text x}+\sqrt[3]{7-\text x}}d\text x \)....equation 1

By property, we know that

I = \(\int\limits_{0}^{7}\cfrac{\sqrt[3]{7-\text x}}{\sqrt[3]{7-\text x}+\sqrt[4]{\text x}}d\text x \)....equation 2

Adding equation 1 and 2

2I = \(\int\limits_{0}^{7}\cfrac{\sqrt[3]{\text x}}{\sqrt[3]{\text x}+\sqrt[3]{7-\text x}}d\text x \) + \(\int\limits_{0}^{7}\cfrac{\sqrt[3]{7-\text x}}{\sqrt[3]{7-\text x}+\sqrt[4]{\text x}}d\text x \)

We know