Find the indefinite integral.∫32x(1+x)dx

An Smith

An Smith

Answered question

2022-05-16

Find the indefinite integral.

32x(1+x)dx

Answer & Explanation

alenahelenash

alenahelenash

Expert2022-06-05Added 556 answers

32x(1+x)dx

Multiply 32x(1+x) by xx.

32x(1+x)xxdx

Combine and simplify the denominator.

3x2(1+x)xdx

Let u=x. Then du=12x12dx, so 2x12du=dx. Rewrite using u and du.

Let u=x. Find dudx.

12x12

Rewrite the problem using u and du.

3u(1+u2)(u2)12du

Simplify.

Multiply the exponents in (u2)12.

3u(1+u2)u1du

Simplify.

3u(1+u2)udu

Cancel the common factor of uu.

31+u2du

Since 3 is constant with respect to u, move 3 out of the integral.

311+u2du

Rewrite 1 as 12.

3112+u2du

The integral of 112+u2 with respect to u is arctan(u)+C.

3(arctan(u)+C)

Simplify.

3arctan(u)+C

Replace all occurrences of u with x.

3arctan(x)+C

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