Seja f(x)=x^2sqrt [3](x+1),xgeqslant -1 Determine a integral indefinida de f(x) a int f(x)dx=(sqrt [3]((x+1)^10))/(10)+(sqrt [3]((x+1)^7))/(7)+(sqrt [3]((x+1)^4))/(4)+c b int f(x)dx=(1)/(4)xsqrt [3]((x+1)^4)+c C. int f(x)dx=(3sqrt [3](x^10))/(10)-(6sqrt [3](x^7))/(7)+(3sqrt [3](x^4))/(4)+c d int f(x)dx=(sqrt [3]((x+1)^10))/(10)-(6sqrt [3]((x+1)^7))/(7)+(sqrt [3]((x+1)^4))/(4)+c e. int f(x)dx=sqrt [3]((x+1)^10)-2sqrt [3]((x+1)^7)+sqrt [3]((x+1)^4)+c

Para determinar a integral indefinida de \( f(x) = x^2 \sqrt[3]{x+1} \) com \( x \geq -1 \), vamos usar a substituição adequada para resolver a integral.<br /><br />Vamos fazer a substituição \( u = x + 1 \). Então, \( du = dx \) e \( x = u - 1 \).<br /><br />Substituindo na integral, temos:<br /><br />\[<br />\int (u-1)^2 \sqrt[3]{u} \, du<br />\]<br /><br />Simplificando, temos:<br /><br />\[<br />\int (u^2 - 2u + 1) \sqrt[3]{u} \, du<br />\]<br /><br />Podemos separar a integral em três partes:<br /><br />\[<br />\int u^2 \sqrt[3]{u} \, du - 2 \int u \sqrt[3]{u} \, du + \int \sqrt[3]{u} \, du<br />\]<br /><br />Vamos calcular cada parte separadamente.<br /><br />Para a primeira parte, \( \int u^2 \sqrt[3]{u} \, du \):<br /><br />\[<br />\int u^2 \sqrt[3]{u} \, du = \int u^{2 + \frac{1}{3}} \, du = \int u^{\frac{7}{3}} \, du = \frac{3}{10} u^{\frac{10}{3}} = \frac{3}{10} (u+1)^{\frac{10}{3}}<br />\]<br /><br />Para a segunda parte, \( -2 \int u \sqrt[3]{u} \, du \):<br /><br />\[<br />-2 \int u \sqrt[3]{u} \, du = -2 \int u^{1 + \frac{1}{3}} \, du = -2 \int u^{\frac{4}{3}} \, du = -2 \cdot \frac{3}{7} u^{\frac{7}{3}} = -\frac{6}{7} (u+1)^{\frac{7}{3}}<br />\]<br /><br />Para a terceira parte, \( \int \sqrt[3]{u} \, du \):<br /><br />\[<br />\int \sqrt[3]{u} \, du = \int u^{\frac{1}{3}} \, du = \frac{3}{4} u^{\frac{4}{3}} = \frac{3}{4} (u+1)^{\frac{4}{3}}<br />\]<br /><br />Somando todas as partes, temos:<br /><br />\[<br />\int f(x) \, dx = \frac{3}{10} (u+1)^{\frac{10}{3}} - \frac{6}{7} (u+1)^{\frac{7}{3}} + \frac{3}{4} (u+1)^{\frac{4}{3}} + c<br />\]<br /><br />Substituindo \( u = x + 1 \), temos:<br /><br />\[<br />\int f(x) \, dx = \frac{3}{10} (x+1)^{\frac{10}{3}} - \frac{6}{7} (x+1)^{\frac{7}{3}} + \frac{3}{4} (x+1)^{\frac{4}{3}} + c<br />\]<br /><br />Portanto, a resposta correta é:<br /><br />d) \(\int f(x) \, dx = \frac{\sqrt[3]{(x+1)^{10}}}{10} - \frac{6\sqrt[3]{(x+1)^{7}}}{7} + \frac{\sqrt[3]{(x+1)^{4}}}{4} + c\)