int frac{2 cos ^{4}(sqrt{x}) sin (sqrt{x})}{3 sqrt{x}} d x

Question

Accepted Answer

Step:1 ：Let u = sqrt{x}, then x = u^2 and dx = 2u duStep:2 ：Substitute u into the integral: int frac{2 cos ^{4}(u) sin (u)}{3 u} (2u) duStep:3 ：Simplify the integral: int 2 cos ^{4}(u) sin (u) duStep:4 ：Integrate with respect to u: frac{1}{12} operatorname{Si}(u) + frac{1}{8} operatorname{Si}(3 u) + frac{1}{24} operatorname{Si}(5 u) + CStep:5 ：Substitute back u = sqrt{x}: frac{1}{12} operatorname{Si}(sqrt{x}) + frac{1}{8} operatorname{Si}(3 sqrt{x}) + frac{1}{24} operatorname{Si}(5 sqrt{x}) + CStep:6 ：(boxed{int frac{2 cos ^{4}(sqrt{x}) sin (sqrt{x})}{3 sqrt{x}} d x = frac{1}{12} operatorname{Si}(sqrt{x}) + frac{1}{8} operatorname{Si}(3 sqrt{x}) + frac{1}{24} operatorname{Si}(5 sqrt{x}) + C})

Answer

Step: 1 ：Let u = sqrt{x}, then x = u^2 and dx = 2u duStep: 2 ：Substitute u into the integral: int frac{2 cos ^{4}(u) sin (u)}{3 u} (2u) duStep: 3 ：Simplify the integral: int 2 cos ^{4}(u) sin (u) duStep: 4 ：Integrate with respect to u: frac{1}{12} operatorname{Si}(u) + frac{1}{8} operatorname{Si}(3 u) + frac{1}{24} operatorname{Si}(5 u) + CStep: 5 ：Substitute back u = sqrt{x}: frac{1}{12} operatorname{Si}(sqrt{x}) + frac{1}{8} operatorname{Si}(3 sqrt{x}) + frac{1}{24} operatorname{Si}(5 sqrt{x}) + CStep: 6 ：(boxed{int frac{2 cos ^{4}(sqrt{x}) sin (sqrt{x})}{3 sqrt{x}} d x = frac{1}{12} operatorname{Si}(sqrt{x}) + frac{1}{8} operatorname{Si}(3 sqrt{x}) + frac{1}{24} operatorname{Si}(5 sqrt{x}) + C})

$\int \frac{2 \cos^{4} (\sqrt{x}) \sin (\sqrt{x})}{3 \sqrt{x}} d x$

Answer

Popular Problems

∫2cos4⁡(x)sin⁡(x)3xdx

Answer

Popular Problems

$\int \frac{2 \cos^{4} (\sqrt{x}) \sin (\sqrt{x})}{3 \sqrt{x}} d x$