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Popular >

integral of xln(x^3+1)

Solution

\int x ln (x^{3} + 1) d x

Solution

\frac{1}{2} x^{2} ln (x^{3} + 1) - \frac{3 x ^{2}}{4} - \frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{4} ln 4 x^{2} - 4 x + 4 + \frac{3}{2} arctan (\frac{1}{3} (2 x - 1)) + C

Solution steps

\int x ln (x^{3} + 1) d x

Apply Integration By Parts

: \frac{1}{2} x^{2} ln (x^{3} + 1) - \int \frac{3 x ^{4}}{2 ( x ^{3} + 1 )} d x

= \frac{1}{2} x^{2} ln (x^{3} + 1) - \int \frac{3 x ^{4}}{2 ( x ^{3} + 1 )} d x

\int \frac{3 x ^{4}}{2 ( x ^{3} + 1 )} d x = \frac{3 x ^{2}}{4} + \frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{4} ln 4 x^{2} - 4 x + 4 - \frac{3}{2} arctan (\frac{2 x - 1}{3})

= \frac{1}{2} x^{2} ln (x^{3} + 1) - (\frac{3 x ^{2}}{4} + \frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{4} ln 4 x^{2} - 4 x + 4 - \frac{3}{2} arctan (\frac{2 x - 1}{3}))

Simplify

\frac{1}{2} x^{2} ln (x^{3} + 1) - (\frac{3 x ^{2}}{4} + \frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{4} ln 4 x^{2} - 4 x + 4 - \frac{3}{2} arctan (\frac{2 x - 1}{3})) : \frac{1}{2} x^{2} ln (x^{3} + 1) - \frac{3 x ^{2}}{4} - \frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{4} ln 4 x^{2} - 4 x + 4 + \frac{3}{2} arctan (\frac{1}{3} (2 x - 1))

= \frac{1}{2} x^{2} ln (x^{3} + 1) - \frac{3 x ^{2}}{4} - \frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{4} ln 4 x^{2} - 4 x + 4 + \frac{3}{2} arctan (\frac{1}{3} (2 x - 1))

Add a constant to the solution

= \frac{1}{2} x^{2} ln (x^{3} + 1) - \frac{3 x ^{2}}{4} - \frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{4} ln 4 x^{2} - 4 x + 4 + \frac{3}{2} arctan (\frac{1}{3} (2 x - 1)) + C

Popular Examples

integral of (x-1)(2x+1)

\int (x - 1) (2 x + 1) d x

integral of x/((2x+5)^2)

\int \frac{x}{( 2 x + 5 ) ^{2}} d x

integral of (4x-2)^5

\int (4 x - 2)^{5} d x

integral of ((x^3+2x))/(sqrt(x^2+4))

\int \frac{( x ^{3} + 2 x )}{x ^{2} + 4} d x

integral of 2^x*2^{2x}

\int 2^{x} \cdot 2^{2 x} d x

Frequently Asked Questions (FAQ)

What is the integral of xln(x^3+1) ?
The integral of xln(x^3+1) is