What is the integral of xcos^2(5x) ?

The integral of xcos^2(5x) is 1/200 (50x^2+10xsin(10x)+cos(10x))+C

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integral of xcos^2(5x)

\int x cos^{2} (5 x) d x

\frac{1}{200} (50 x^{2} + 10 x sin (10 x) + cos (10 x)) + C

Solution steps

\int x cos^{2} (5 x) d x

Rewrite using trig identities

= \int \frac{1}{2} x (1 + cos (10 x)) d x

Take the constant out:

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

= \frac{1}{2} \cdot \int x (1 + cos (10 x)) d x

Apply Integration By Parts

: x (x + \frac{1}{10} sin (10 x)) - \int x + \frac{1}{10} sin (10 x) d x

= \frac{1}{2} (x (x + \frac{1}{10} sin (10 x)) - \int x + \frac{1}{10} sin (10 x) d x)

\int x + \frac{1}{10} sin (10 x) d x = \frac{x ^{2}}{2} - \frac{1}{100} cos (10 x)

= \frac{1}{2} (x (x + \frac{1}{10} sin (10 x)) - (\frac{x ^{2}}{2} - \frac{1}{100} cos (10 x)))

Simplify

\frac{1}{2} (x (x + \frac{1}{10} sin (10 x)) - (\frac{x ^{2}}{2} - \frac{1}{100} cos (10 x))) : \frac{1}{200} (50 x^{2} + 10 x sin (10 x) + cos (10 x))

= \frac{1}{200} (50 x^{2} + 10 x sin (10 x) + cos (10 x))

Add a constant to the solution

= \frac{1}{200} (50 x^{2} + 10 x sin (10 x) + cos (10 x)) + C

x \to - 3 + lim (\frac{x ^{3}}{x ^{2} - 9})

y = tan (e^{x})

\int_{3}^{4} x^{2} d x

\frac{d}{d x} (\frac{5 x + 1}{2 x})

e^{3 - t}

What is the integral of xcos^2(5x) ?
The integral of xcos^2(5x) is 1/200 (50x^2+10xsin(10x)+cos(10x))+C