What is the integral of (xsin(x))/(cos^3(x)) ?

The integral of (xsin(x))/(cos^3(x)) is 1/2 xtan^2(x)-1/2 (-x+tan(x))+C

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integral of (xsin(x))/(cos^3(x))

\int \frac{x sin ( x )}{cos ^{3} ( x )} d x

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

Solution steps

\int \frac{x sin ( x )}{cos ^{3} ( x )} d x

Rewrite using trig identities

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

Apply Integration By Parts

: \frac{1}{2} x tan^{2} (x) - \int \frac{1}{2} tan^{2} (x) d x

= \frac{1}{2} x tan^{2} (x) - \int \frac{1}{2} tan^{2} (x) d x

\int \frac{1}{2} tan^{2} (x) d x = \frac{1}{2} (- x + tan (x))

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

Add a constant to the solution

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

\int \frac{1 + ln ( x )}{x ln ( x )} d x

\int x e^{x} + x d x

\int \frac{x e ^{x}}{( 1 + e ^{x} ) ^{2}} d x

\int (e^{2 x} + \frac{1}{1 - x}) d x

\int \frac{x - 2}{x ^{2} + 1} d x

What is the integral of (xsin(x))/(cos^3(x)) ?
The integral of (xsin(x))/(cos^3(x)) is 1/2 xtan^2(x)-1/2 (-x+tan(x))+C