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261.

Evaluate: $\int_{0}^{π} \frac{xsinx}{1 + \cos^{2} x} dx$

262.

Evaluate: $\int_{- a}^{a} \sqrt{\frac{a - x}{a + x}} dx$

263.

Evaluate: $\int \sec^{2} (7 - x) dx$

264.

If $\int_{0}^{1} (3 x^{2} + 2 x + k) dx = 0$ , find the value of k.

265.

Evaluate: $\int \frac{e^{x}}{\sqrt{5 - 4 e^{x} - e^{2 x}}} dx$

266.

Evaluate: $\int \frac{(x - 4) e^{x}}{{(x - 2)}^{3}} dx .$

267.

Evaluate: $\int_{0}^{π} \frac{e^{cosx}}{e^{cosx} + e^{- cosx}}$

268.

Evaluate: $\int_{0}^{\frac{π}{2}} (2 \log \sin x - \log \sin 2 x) dx$

$I = \int \frac{\log x}{x} dx$

Substituting log x = t

Differentiating w.r.t. x, we get

$\frac{1}{x} d x = d t$

On substitution, we get

$I = \int \frac{\log x}{x} dx = \int t . dt = \frac{t^{2}}{2} + C = \frac{{(\log x)}^{2}}{2} + C$

270.

Evaluate: $\int_{0}^{π} \frac{x}{1 + sinx} dx$