How do you find the integral of $\int x cos (5 x) d x$ $intxcos(5x)dx$ ?

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How do you find the integral of $\int x cos (5 x) d x$ $intxcos(5x)dx$ ?

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Answer 1 · 2017-03-23T15:58:41

Use integration by parts.
$\int u d v = u v - \int v d u$ $intudv=uv-intvdu$

Explanation:

Let $u = x$ $u = x$ , then $d u = d x, d v = cos (5 x) d x, and v = \frac{1}{5} sin (5 x)$ $du = dx, dv=cos(5x)dx, and v=1/5sin(5x)$

Substituting these values into the integration by parts equation:

$\int x cos (5 x) d x = \frac{x}{5} sin (5 x) - \frac{1}{5} \int sin (5 x) d x$ $intxcos(5x)dx=x/5sin(5x)-1/5intsin(5x)dx$

The remaining integral on the right is trivial:

$\int x cos (5 x) d x = \frac{x}{5} sin (5 x) + \frac{1}{25} cos (5 x) + C$ $intxcos(5x)dx=x/5sin(5x)+1/25cos(5x)+C$

Answer 2 · 2017-03-23T16:20:52

$\frac{1}{25} (5 x sin 5 x + cos 5 x) + C$ $1/25(5x sin 5x + cos 5x) + C$

Explanation:

Use integration by parts $\int u d v = u \cdot v - \int v d u$ $int u dv = u*v - int v du$

Let $u = x, d u = d x$ $u = x, du = dx$
Let $d v = cos 5 x d x, v = \int cos 5 x d x$ $dv = cos 5x dx, v = int cos 5x dx$

Let $w = 5 x, d w = 5 d x$ $w = 5x, dw = 5 dx$
$\int cos 5 x d x = \frac{1}{5} \int cos w d w = \frac{1}{5} sin 5 x$ $int cos 5x dx = 1/5 int cos w dw = 1/5 sin 5x$

Fill in the integration by parts:
$\int x cos 5 x d x = \frac{1}{5} x sin 5 x - \frac{1}{5} \int sin 5 x d x$ $int x cos 5x dx = 1/5x sin 5x - 1/5 int sin 5xdx$

To complete the last integration piece:
Let $w = 5 x, d w = 5 d x$ $w = 5x, dw = 5 dx$

$- \frac{1}{5} \int sin 5 x d x = - \frac{1}{5} \cdot \frac{1}{5} \int 5 sin 5 x d x = - \frac{1}{25} \int sin w d w = - \frac{1}{25} (- cos 5 x) = \frac{1}{25} cos 5 x$ $-1/5int sin 5xdx = -1/5 *1/5 int 5sin 5x dx = -1/25 int sin w dw = -1/25 (-cos 5x) = 1/25 cos 5x$

The final solution simplified:
$\int x cos 5 x d x = \frac{1}{5} x sin 5 x + \frac{1}{25} cos 5 x + C = \frac{1}{25} (5 x sin 5 x + cos 5 x) + C$ $int x cos 5x dx = 1/5x sin 5x +1/25 cos 5x + C= 1/25(5x sin 5x + cos 5x) + C$

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How do you find the integral of $\int x cos (5 x) d x$ $intxcos(5x)dx$ ?

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