What is the integral of cos(xy^2) ?

The integral of cos(xy^2) is 1/(y^2)sin(y^2x)+C

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integral of cos(xy^2)

\int cos (x y^{2}) d x

\frac{1}{y ^{2}} sin (y^{2} x) + C

Solution steps

\int cos (x y^{2}) d x

Apply u-substitution

: \int \frac{cos ( u )}{y ^{2}} d u

= \int \frac{cos ( u )}{y ^{2}} d u

Take the constant out:

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

= \frac{1}{y ^{2}} \cdot \int cos (u) d u

Use the common integral:

\int cos (u) d u = sin (u)

= \frac{1}{y ^{2}} sin (u)

Substitute back

u = y^{2} x

= \frac{1}{y ^{2}} sin (y^{2} x)

Add a constant to the solution

= \frac{1}{y ^{2}} sin (y^{2} x) + C

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What is the integral of cos(xy^2) ?
The integral of cos(xy^2) is 1/(y^2)sin(y^2x)+C