Question

integration x(x+1)^-1​

Answer 1

Answer:

$\quad \dashrightarrow \: \: \bf\frac{x^{3}}{3}+\frac{x^{2}}{2}+С$

Step-by-step Explanation :

$\quad \maltese \: \:\displaystyle \bf\int{ x(x+1) ^ { 1 } }d x$

$\quad \dashrightarrow \: \: \tt\int x\left(x+1\right)\mathrm{d}x$

$\quad \dashrightarrow \: \: \tt\int x^{2}+x\mathrm{d}x$

$\quad \dashrightarrow \: \: \tt \int x^{2}\mathrm{d}x+\int x\mathrm{d}x$

$\quad \dashrightarrow \: \: \tt \frac{x^{3}}{3}+\int x\mathrm{d}x$

$\quad \dashrightarrow \: \: \tt \frac{x^{3}}{3}+\frac{x^{2}}{2}$

$\quad \dashrightarrow \: \: \bf \frac{x^{3}}{3}+\frac{x^{2}}{2}+С \: \: \bigstar$

Method Used :

• Calculate x+1 to the power of 1 and get x+1.

• Use the distributive property to multiply x by x+1.

• Integrate the sum term by term.

• Since int x^{k} dx = {x^{k+1}}/{k+1} for k ≠ -1, replace int x^{2} dx with {x^{3}}/{3}.

• Since int x^{k} dx = {x^{k+1}}/{k+1} for k ≠ -1, replace int x dx with {x^{2}}/{2}.

• If F (x) is an antiderivative of f (x), then the set of all antiderivatives of f (x) is given by F (x)+C. Therefore, add the constant of integration C € R to the result.