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Câu hỏi: Biết $\int{f\left( 2x \right)\text{d}x}={{\sin }^{2}}x+\ln x+C$. Tìm nguyên hàm $\int{f\left( x \right)\text{d}x}$ ?
A. $\int{f\left( x \right)\text{d}x}={{\sin }^{2}}\dfrac{x}{2}+\ln x+C$.
B. $\int{f\left( x \right)\text{d}x}=2{{\sin }^{2}}2x+2\ln x+C$.
C. $\int{f\left( x \right)\text{d}x}=2{{\sin }^{2}}\dfrac{x}{2}+2\ln x+C$.
D. $\int{f\left( x \right)\text{d}x}=2{{\sin }^{2}}x+2\ln x+C$.
Ta có: $\int{f\left( 2x \right)\text{d}x}={{\sin }^{2}}x+\ln x+C\Leftrightarrow \dfrac{1}{2}\int{f\left( 2x \right)\text{d}\left( 2x \right)}=\dfrac{1-\cos 2x}{2}+\ln \left( 2x \right)-\ln 2+C$
$\Leftrightarrow $ $\int{f\left( 2x \right)\text{d}\left( 2x \right)}=1-\cos 2x+2\ln \left( 2x \right)-2\ln 2+2C$
$\Leftrightarrow \int{f\left( x \right)\text{d}x}=1-\cos x+2\ln x-2\ln 2+2C\Leftrightarrow \int{f\left( x \right)\text{d}x}=2{{\sin }^{2}}\dfrac{x}{2}+2\ln x+{C}'$.
A. $\int{f\left( x \right)\text{d}x}={{\sin }^{2}}\dfrac{x}{2}+\ln x+C$.
B. $\int{f\left( x \right)\text{d}x}=2{{\sin }^{2}}2x+2\ln x+C$.
C. $\int{f\left( x \right)\text{d}x}=2{{\sin }^{2}}\dfrac{x}{2}+2\ln x+C$.
D. $\int{f\left( x \right)\text{d}x}=2{{\sin }^{2}}x+2\ln x+C$.
Ta có: $\int{f\left( 2x \right)\text{d}x}={{\sin }^{2}}x+\ln x+C\Leftrightarrow \dfrac{1}{2}\int{f\left( 2x \right)\text{d}\left( 2x \right)}=\dfrac{1-\cos 2x}{2}+\ln \left( 2x \right)-\ln 2+C$
$\Leftrightarrow $ $\int{f\left( 2x \right)\text{d}\left( 2x \right)}=1-\cos 2x+2\ln \left( 2x \right)-2\ln 2+2C$
$\Leftrightarrow \int{f\left( x \right)\text{d}x}=1-\cos x+2\ln x-2\ln 2+2C\Leftrightarrow \int{f\left( x \right)\text{d}x}=2{{\sin }^{2}}\dfrac{x}{2}+2\ln x+{C}'$.
Đáp án C.