What is the integral of 2x^2sec^2(x^3+1) ?

The integral of 2x^2sec^2(x^3+1) is 2/3 tan(x^3+1)+C

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integral of 2x^2sec^2(x^3+1)

\int 2 x^{2} sec^{2} (x^{3} + 1) d x

\frac{2}{3} tan (x^{3} + 1) + C

Solution steps

\int 2 x^{2} sec^{2} (x^{3} + 1) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int x^{2} sec^{2} (x^{3} + 1) d x

Apply u-substitution

: \int \frac{sec ^{2} ( u )}{3} d u

= 2 \cdot \int \frac{sec ^{2} ( u )}{3} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \frac{1}{3} \cdot \int sec^{2} (u) d u

Use the common integral:

\int sec^{2} (u) d u = tan (u)

= 2 \cdot \frac{1}{3} tan (u)

Substitute back

u = x^{3} + 1

= 2 \cdot \frac{1}{3} tan (x^{3} + 1)

Simplify

2 \cdot \frac{1}{3} tan (x^{3} + 1) : \frac{2}{3} tan (x^{3} + 1)

= \frac{2}{3} tan (x^{3} + 1)

Add a constant to the solution

= \frac{2}{3} tan (x^{3} + 1) + C

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What is the integral of 2x^2sec^2(x^3+1) ?
The integral of 2x^2sec^2(x^3+1) is 2/3 tan(x^3+1)+C